PunjabiUni

ਸਿੱਖਣ ਦੇ ਉਦੇਸ਼

ਇਸ ਭਾਗ ਦੇ ਅੰਤ ਤੱਕ, ਤੁਸੀਂ ਯੋਗ ਹੋਵੋਗੇ:

ਬਹੁਪਦਾਂ ਦੀ ਡਿਗਰੀ ਨਿਰਧਾਰਤ ਕਰੋ

ਬਹੁਪਦਾਂ ਨੂੰ ਜੋੜੋ ਅਤੇ ਘਟਾਓ

ਦਿੱਤੇ ਗਏ ਮੁੱਲ ਲਈ ਬਹੁਪਦੀ ਫੰਕਸ਼ਨ ਦਾ ਮੁਲਾਂਕਣ ਕਰੋ

ਬਹੁਪਦੀ ਫੰਕਸ਼ਨਾਂ ਨੂੰ ਜੋੜੋ ਅਤੇ ਘਟਾਓ

ਤਿਆਰ ਰਹੋ 5.1

ਸ਼ੁਰੂ ਕਰਨ ਤੋਂ ਪਹਿਲਾਂ, ਇਸ ਤਿਆਰੀ ਕਵਿਜ਼ ਨੂੰ ਲਓ।

ਸਰਲ ਕਰੋ: 3x2+3x+1+8x2+5x+5.3x2+3x+1+8x2+5x+5. ਜੇਕਰ ਤੁਸੀਂ ਇਹ ਸਮੱਸਿਆ ਗੁਆ ਦਿੱਤੀ ਹੈ, ਤਾਂ ਉਦਾਹਰਨ 1.7 ਦੀ ਸਮੀਖਿਆ ਕਰੋ।

ਤਿਆਰ ਰਹੋ 5.2

ਘਟਾਓ: (5n+8)−(2n−1).(5n+8)−(2n−1). ਜੇਕਰ ਤੁਸੀਂ ਇਹ ਸਮੱਸਿਆ ਗੁਆ ਦਿੱਤੀ ਹੈ, ਤਾਂ ਉਦਾਹਰਨ 1.5 ਦੀ ਸਮੀਖਿਆ ਕਰੋ।

ਤਿਆਰ ਰਹੋ 5.3

ਮੁਲਾਂਕਣ ਕਰੋ: 4xy24xy2 ਜਦੋਂ x=−2x=−2 ਅਤੇ y=5.y=5. ਜੇਕਰ ਤੁਸੀਂ ਇਹ ਸਮੱਸਿਆ ਗੁਆ ਦਿੱਤੀ ਹੈ, ਤਾਂ ਉਦਾਹਰਨ 1.21 ਦੀ ਸਮੀਖਿਆ ਕਰੋ।

ਬਹੁਪਦਾਂ ਦੀ ਡਿਗਰੀ ਨਿਰਧਾਰਤ ਕਰੋ

ਅਸੀਂ ਸਿੱਖਿਆ ਹੈ ਕਿ ਇੱਕ ਪਦ ਇੱਕ ਸਥਿਰ ਅੰਕ ਜਾਂ ਇੱਕ ਸਥਿਰ ਅੰਕ ਅਤੇ ਇੱਕ ਜਾਂ ਇੱਕ ਤੋਂ ਵੱਧ ਚਲਾਂ ਦਾ ਗੁਣਨਫਲ ਹੁੰਦਾ ਹੈ। ਇੱਕ ਮੋਨੋਮੀਅਲ ਇੱਕ ਪਦ ਵਾਲਾ ਇੱਕ ਬੀਜਗਣਿਤਿਕ ਸਮੀਕਰਨ ਹੈ। ਜਦੋਂ ਇਹ axm,axm ਦੇ ਰੂਪ ਵਿੱਚ ਹੁੰਦਾ ਹੈ, ਜਿੱਥੇ a ਇੱਕ ਸਥਿਰ ਅੰਕ ਹੈ ਅਤੇ m ਇੱਕ ਪੂਰਨ ਸੰਖਿਆ ਹੈ, ਇਸਨੂੰ ਇੱਕ ਚਲ ਵਿੱਚ ਇੱਕ ਮੋਨੋਮੀਅਲ ਕਿਹਾ ਜਾਂਦਾ ਹੈ। ਇੱਕ ਚਲ ਵਿੱਚ ਮੋਨੋਮੀਅਲ ਦੇ ਕੁਝ ਉਦਾਹਰਨ 2x, 5y, 17z, 2x,5y,17z, ਅਤੇ 4y2 4y2 ਹਨ। ਮੋਨੋਮੀਅਲ ਵਿੱਚ ਇੱਕ ਤੋਂ ਵੱਧ ਚਲ ਵੀ ਹੋ ਸਕਦੇ ਹਨ ਜਿਵੇਂ ਕਿ 5abc5abc ਅਤੇ −4a2b3c2.−4a2b3c2.

ਮੋਨੋਮੀਅਲ

ਮੋਨੋਮੀਅਲ ਇੱਕ ਬੀਜਗਣਿਤਿਕ ਸਮੀਕਰਨ ਹੈ ਜਿਸ ਵਿੱਚ ਇੱਕ ਪਦ ਹੁੰਦਾ ਹੈ।

ਇੱਕ ਚਲ ਵਿੱਚ ਮੋਨੋਮੀਅਲ axm,axm ਦਾ ਇੱਕ ਪਦ ਹੈ, ਜਿੱਥੇ a ਇੱਕ ਸਥਿਰ ਅੰਕ ਹੈ ਅਤੇ m ਇੱਕ ਪੂਰਨ ਸੰਖਿਆ ਹੈ।

ਇੱਕ ਮੋਨੋਮੀਅਲ, ਜਾਂ ਜੋੜ ਜਾਂ ਘਟਾਓ ਦੁਆਰਾ ਜੋੜੇ ਗਏ ਦੋ ਜਾਂ ਦੋ ਤੋਂ ਵੱਧ ਮੋਨੋਮੀਅਲ, ਇੱਕ ਬਹੁਪਦ ਹੁੰਦਾ ਹੈ। ਕੁਝ ਬਹੁਪਦਾਂ ਦੇ ਵਿਸ਼ੇਸ਼ ਨਾਮ ਹੁੰਦੇ ਹਨ, ਜੋ ਪਦਾਂ ਦੀ ਗਿਣਤੀ 'ਤੇ ਅਧਾਰਤ ਹੁੰਦੇ ਹਨ। ਇੱਕ ਮੋਨੋਮੀਅਲ ਬਿਲਕੁਲ ਇੱਕ ਪਦ ਵਾਲਾ ਬਹੁਪਦ ਹੁੰਦਾ ਹੈ। ਇੱਕ ਬਾਈਨੋਮੀਅਲ ਵਿੱਚ ਬਿਲਕੁਲ ਦੋ ਪਦ ਹੁੰਦੇ ਹਨ, ਅਤੇ ਇੱਕ ਟ੍ਰਾਈਨੋਮੀਅਲ ਵਿੱਚ ਬਿਲਕੁਲ ਤਿੰਨ ਪਦ ਹੁੰਦੇ ਹਨ। ਤਿੰਨ ਤੋਂ ਵੱਧ ਪਦਾਂ ਵਾਲੇ ਬਹੁਪਦਾਂ ਲਈ ਕੋਈ ਵਿਸ਼ੇਸ਼ ਨਾਮ ਨਹੀਂ ਹਨ।

ਬਹੁਪਦ

ਬਹੁਪਦ—ਇੱਕ ਮੋਨੋਮੀਅਲ, ਜਾਂ ਜੋੜ ਜਾਂ ਘਟਾਓ ਦੁਆਰਾ ਜੋੜੇ ਗਏ ਦੋ ਜਾਂ ਦੋ ਤੋਂ ਵੱਧ ਬੀਜਗਣਿਤਿਕ ਪਦ ਇੱਕ ਬਹੁਪਦ ਹੁੰਦੇ ਹਨ।

ਮੋਨੋਮੀਅਲ—ਬਿਲਕੁਲ ਇੱਕ ਪਦ ਵਾਲੇ ਬਹੁਪਦ ਨੂੰ ਮੋਨੋਮੀਅਲ ਕਿਹਾ ਜਾਂਦਾ ਹੈ।

ਬਾਈਨੋਮੀਅਲ—ਬਿਲਕੁਲ ਦੋ ਪਦਾਂ ਵਾਲੇ ਬਹੁਪਦ ਨੂੰ ਬਾਈਨੋਮੀਅਲ ਕਿਹਾ ਜਾਂਦਾ ਹੈ।

ਟ੍ਰਾਈਨੋਮੀਅਲ—ਬਿਲਕੁਲ ਤਿੰਨ ਪਦਾਂ ਵਾਲੇ ਬਹੁਪਦ ਨੂੰ ਟ੍ਰਾਈਨੋਮੀਅਲ ਕਿਹਾ ਜਾਂਦਾ ਹੈ।

Here are some examples of polynomials.

row: Polynomial | y+1y+1 | 4a2−7ab+2b24a2−7ab+2b2 | 4x4+x3+8x2−9x+14x4+x3+8x2−9x+1

row: Monomial | 14 | 8y28y2 | −9x3y5−9x3y5 | −13a3b2c−13a3b2c

row: Binomial | a+7ba+7b | 4x2−y24x2−y2 | y2−16y2−16 | 3p3q−9p2q3p3q−9p2q

row: Trinomial | x2−7x+12x2−7x+12 | 9m2+2mn−8n29m2+2mn−8n2 | 6k4−k3+8k6k4−k3+8k | z4+3z2−1z4+3z2−1

Notice that every monomial, binomial, and trinomial is also a polynomial. They are just special members of the “family” of polynomials and so they have special names. We use the words monomial, binomial, and trinomial when referring to these special polynomials and just call all the rest polynomials.

The degree of a polynomial and the degree of its terms are determined by the exponents of the variable.

A monomial that has no variable, just a constant, is a special case. The degree of a constant is 0.

Degree of a Polynomial

The degree of a term is the sum of the exponents of its variables.

The degree of a constant is 0.

The degree of a polynomial is the highest degree of all its terms.

Let’s see how this works by looking at several polynomials. We’ll take it step by step, starting with monomials, and then progressing to polynomials with more terms.

Let's start by looking at a monomial. The monomial 8ab28ab2 has two variables a and b. To find the degree we need to find the sum of the exponents. The variable a doesn't have an exponent written, but remember that means the exponent is 1. The exponent of b is 2. The sum of the exponents, 1+2,1+2, is 3 so the degree is 3.

Here are some additional examples.

Working with polynomials is easier when you list the terms in descending order of degrees. When a polynomial is written this way, it is said to be in standard form of a polynomial. Get in the habit of writing the term with the highest degree first.

Example 5.1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

ⓐ 7y2−5y+37y2−5y+3 ⓑ −2a4b2−2a4b2 ⓒ 3x5−4x3−6x2+x−83x5−4x3−6x2+x−8 ⓓ 2y−8xy32y−8xy3 ⓔ 15

Solution

row: Polynomial | Number of terms | Type | Degree of terms | Degree of polynomial

row: ⓐ | 7y2−5y+37y2−5y+3 | 3 | Trinomial | 2, 1, 0 | 2

row: ⓑ | −2a4b2−2a4b2 | 1 | Monomial | 6 | 6

row: ⓒ | 3x5−4x3−6x2+x−83x5−4x3−6x2+x−8 | 5 | Polynomial | 5, 3, 2, 1, 0 | 5

row: ⓓ | 2y−8xy32y−8xy3 | 2 | Binomial | 1, 4 | 4

row: ⓔ | 15 | 1 | Monomial | 0 | 0

Try It 5.1

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

ⓐ −5−5 ⓑ 8y3−7y2−y−38y3−7y2−y−3 ⓒ −3x2y−5xy+9xy3−3x2y−5xy+9xy3 ⓓ 81m2−4n281m2−4n2 ⓔ −3x6y3z−3x6y3z

Try It 5.2

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

ⓐ 64k3−864k3−8 ⓑ 9m3+4m2−29m3+4m2−2 ⓒ 5656 ⓓ 8a4−7a3b−6a2b2−4ab3+7b48a4−7a3b−6a2b2−4ab3+7b4 ⓔ −p4q3−p4q3

Add and Subtract Polynomials

We have learned how to simplify expressions by combining like terms. Remember, like terms must have the same variables with the same exponent. Since monomials are terms, adding and subtracting monomials is the same as combining like terms. If the monomials are like terms, we just combine them by adding or subtracting the coefficients.

Example 5.2

Add or subtract: ⓐ 25y2+15y225y2+15y2 ⓑ 16pq3−(−7pq3).16pq3−(−7pq3).

Solution

ⓐ 25y2+15y2 Combine like terms.40y2 25y2+15y2 Combine like terms.40y2

ⓑ 16pq3−(−7pq3) Combine like terms.23pq3 16pq3−(−7pq3) Combine like terms.23pq3

Try It 5.3

Add or subtract: ⓐ 12q2+9q212q2+9q2 ⓑ 8mn3−(−5mn3).8mn3−(−5mn3).

Try It 5.4

Add or subtract: ⓐ −15c2+8c2−15c2+8c2 ⓑ −15y2z3−(−5y2z3).−15y2z3−(−5y2z3).

Remember that like terms must have the same variables with the same exponents.

Example 5.3

Simplify: ⓐ a2+7b2−6a2a2+7b2−6a2 ⓑ u2v+5u2−3v2.u2v+5u2−3v2.

Solution

ⓐ

row: a2+7b2−6a2a2+7b2−6a2

row: Combine like terms. | −5a2+7b2−5a2+7b2

ⓑ

row: u2v+5u2−3v2u2v+5u2−3v2

row: There are no like terms to combine.In this case, the polynomial is unchanged. | u2v+5u2−3v2u2v+5u2−3v2

Try It 5.5

Add: ⓐ 8y2+3z2−3y28y2+3z2−3y2 ⓑ m2n2−8m2+4n2.m2n2−8m2+4n2.

Try It 5.6

Add: ⓐ 3m2+n2−7m23m2+n2−7m2 ⓑ pq2−6p−5q2.pq2−6p−5q2.

We can think of adding and subtracting polynomials as just adding and subtracting a series of monomials. Look for the like terms—those with the same variables and the same exponent. The Commutative Property allows us to rearrange the terms to put like terms together.

Example 5.4

Find the sum:(7y2−2y+9)+(4y2−8y−7).(7y2−2y+9)+(4y2−8y−7).

Solution

row: Identify like terms. | (7y2________−2y___+9)+(4y2________−8y___−7)(7y2________−2y___+9)+(4y2________−8y___−7)

row: Rewrite without the parentheses,rearranging to get the like terms together. | 7y2+4y2__________________−2y−8y_______+9−77y2+4y2__________________−2y−8y_______+9−7

row: Combine like terms. | 11y2−10y+211y2−10y+2

Try It 5.7

Find the sum: (7x2−4x+5)+(x2−7x+3).(7x2−4x+5)+(x2−7x+3).

Try It 5.8

Find the sum: (14y2+6y−4)+(3y2+8y+5).(14y2+6y−4)+(3y2+8y+5).

Be careful with the signs as you distribute while subtracting the polynomials in the next example.

Example 5.5

Find the difference: (9w2−7w+5)−(2w2−4).(9w2−7w+5)−(2w2−4).

Solution

row: (9w2−7w+5)−(2w2−4)(9w2−7w+5)−(2w2−4)

row: Distribute and identify like terms. | 9w2________−7w___+5−2w2________+49w2________−7w___+5−2w2________+4

row: Rearrange the terms. | 9w2−2w2____________________−7w___+5+49w2−2w2____________________−7w___+5+4

row: Combine like terms. | 7w2−7w+97w2−7w+9

Try It 5.9

Find the difference: (8x2+3x−19)−(7x2−14).(8x2+3x−19)−(7x2−14).

Try It 5.10

Find the difference: (9b2−5b−4)−(3b2−5b−7).(9b2−5b−4)−(3b2−5b−7).

To subtract aa from b,b, we write it as b−a,b−a, placing the bb first.

Example 5.6

Subtract (p2+10pq−2q2)(p2+10pq−2q2) from (p2+q2).(p2+q2).

Solution

row: (p2+q2)−(p2+10pq−2q2)(p2+q2)−(p2+10pq−2q2)

row: Distribute. | p2+q2−p2−10pq+2q2p2+q2−p2−10pq+2q2

row: Rearrange the terms, to put like terms together. | p2−p2−10pq+q2+2q2p2−p2−10pq+q2+2q2

row: Combine like terms. | −10pq+3q2−10pq+3q2

Try It 5.11

Subtract (a2+5ab−6b2)(a2+5ab−6b2) from (a2+b2).(a2+b2).

Try It 5.12

Subtract (m2−7mn−3n2)(m2−7mn−3n2) from (m2+n2).(m2+n2).

Example 5.7

Find the sum: (u2−6uv+5v2)+(3u2+2uv).(u2−6uv+5v2)+(3u2+2uv).

Solution

row: (u2−6uv+5v2)+(3u2+2uv)(u2−6uv+5v2)+(3u2+2uv)

row: Distribute. | u2−6uv+5v2+3u2+2uvu2−6uv+5v2+3u2+2uv

row: Rearrange the terms to put like terms together. | u2+3u2−6uv+2uv+5v2u2+3u2−6uv+2uv+5v2

row: Combine like terms. | 4u2−4uv+5v24u2−4uv+5v2

Try It 5.13

Find the sum: (3x2−4xy+5y2)+(2x2−xy).(3x2−4xy+5y2)+(2x2−xy).

Try It 5.14

Find the sum: (2x2−3xy−2y2)+(5x2−3xy).(2x2−3xy−2y2)+(5x2−3xy).

When we add and subtract more than two polynomials, the process is the same.

Example 5.8

Simplify: (a3−a2b)−(ab2+b3)+(a2b+ab2).(a3−a2b)−(ab2+b3)+(a2b+ab2).

Solution

row: (a3−a2b)−(ab2+b3)+(a2b+ab2)(a3−a2b)−(ab2+b3)+(a2b+ab2)

row: Distribute. | a3−a2b−ab2−b3+a2b+ab2a3−a2b−ab2−b3+a2b+ab2

row: Rewrite without the parentheses,rearranging to get the like terms together. | a3−a2b+a2b−ab2+ab2−b3a3−a2b+a2b−ab2+ab2−b3

row: Combine like terms. | a3−b3a3−b3

Try It 5.15

Simplify: (x3−x2y)−(xy2+y3)+(x2y+xy2).(x3−x2y)−(xy2+y3)+(x2y+xy2).

Try It 5.16

Simplify: (p3−p2q)+(pq2+q3)−(p2q+pq2).(p3−p2q)+(pq2+q3)−(p2q+pq2).

Evaluate a Polynomial Function for a Given Value

A polynomial function is a function defined by a polynomial. For example, f(x)=x2+5x+6f(x)=x2+5x+6 and g(x)=3x−4g(x)=3x−4 are polynomial functions, because x2+5x+6x2+5x+6 and 3x−43x−4 are polynomials.

Polynomial Function

A polynomial function is a function whose range values are defined by a polynomial.

In Graphs and Functions, where we first introduced functions, we learned that evaluating a function means to find the value of f(x)f(x) for a given value of x. To evaluate a polynomial function, we will substitute the given value for the variable and then simplify using the order of operations.

Example 5.9

For the function f(x)=5x2−8x+4f(x)=5x2−8x+4 find: ⓐ f(4)f(4) ⓑ f(−2)f(−2) ⓒ f(0).f(0).

Solution

ⓐ

row: Simplify the exponents.

row: Multiply.

row: Simplify.

ⓑ

row: Simplify the exponents.

row: Multiply.

row: Simplify.

ⓒ

row: Simplify the exponents.

row: Multiply.

row: Simplify.

Try It 5.17

For the function f(x)=3x2+2x−15,f(x)=3x2+2x−15, find ⓐ f(3)f(3) ⓑ f(−5)f(−5) ⓒ f(0).f(0).

Try It 5.18

For the function g(x)=5x2−x−4,g(x)=5x2−x−4, find ⓐ g(−2)g(−2) ⓑ g(−1)g(−1) ⓒ g(0).g(0).

The polynomial functions similar to the one in the next example are used in many fields to determine the height of an object at some time after it is projected into the air. The polynomial in the next function is used specifically for dropping something from 250 ft.

Example 5.10

The polynomial function h(t)=−16t2+250h(t)=−16t2+250 gives the height of a ball t seconds after it is dropped from a 250-foot tall building. Find the height after t=2t=2 seconds.

Solution

row: h(t)=−16t2+250h(t)=−16t2+250

row: To find h(2),h(2), substitute t=2.t=2. | h(2)=−16(2)2+250h(2)=−16(2)2+250

row: Simplify. | h(2)=−16·4+250h(2)=−16·4+250

row: Simplify. | h(2)=−64+250h(2)=−64+250

row: Simplify. | h(2)=186h(2)=186

row: After 2 seconds the height of the ball is 186 feet.

Try It 5.19

The polynomial function h(t)=−16t2+150h(t)=−16t2+150 gives the height of a stone t seconds after it is dropped from a 150-foot tall cliff. Find the height after t=0t=0 seconds (the initial height of the object).

Try It 5.20

The polynomial function h(t)=−16t2+175h(t)=−16t2+175 gives the height of a ball t seconds after it is dropped from a 175-foot tall bridge. Find the height after t=3t=3 seconds.

Add and Subtract Polynomial Functions

Just as polynomials can be added and subtracted, polynomial functions can also be added and subtracted.

Addition and Subtraction of Polynomial Functions

For functions f(x)f(x) and g(x),g(x),

Example 5.11

For functions f(x)=3x2−5x+7f(x)=3x2−5x+7 and g(x)=x2−4x−3,g(x)=x2−4x−3, find:

ⓐ (f+g)(x)(f+g)(x) ⓑ (f+g)(3)(f+g)(3) ⓒ (f−g)(x)(f−g)(x) ⓓ (f−g)(−2).(f−g)(−2).

Solution

ⓐ

row: Rewrite without the parentheses.

row: Put like terms together.

row: Combine like terms.

ⓑ In part (a) we found (f+g)(x)(f+g)(x) and now are asked to find (f+g)(3).(f+g)(3).

row: (f+g)(x)=4x2−9x+4(f+g)(x)=4x2−9x+4

row: To find (f+g)(3),(f+g)(3), substitute x=3.x=3. | (f+g)(3)=4(3)2−9·3+4(f+g)(3)=4(3)2−9·3+4

row: (f+g)(3)=4·9−9·3+4(f+g)(3)=4·9−9·3+4

row: (f+g)(3)=36−27+4(f+g)(3)=36−27+4

Notice that we could have found (f+g)(3)(f+g)(3) by first finding the values of f(3)f(3) and g(3)g(3) separately and then adding the results.

row: Find f(3).f(3).

row: Find g(3).g(3).

row: Find (f+g)(3).(f+g)(3).

ⓒ

row: Rewrite without the parentheses.

row: Put like terms together.

row: Combine like terms.

ⓓ

Try It 5.21

For functions f(x)=2x2−4x+3f(x)=2x2−4x+3 and g(x)=x2−2x−6,g(x)=x2−2x−6, find: ⓐ (f+g)(x)(f+g)(x) ⓑ (f+g)(3)(f+g)(3) ⓒ (f−g)(x)(f−g)(x) ⓓ (f−g)(−2).(f−g)(−2).

Try It 5.22

For functions f(x)=5x2−4x−1f(x)=5x2−4x−1 and g(x)=x2+3x+8,g(x)=x2+3x+8, find ⓐ (f+g)(x)(f+g)(x) ⓑ (f+g)(3)(f+g)(3) ⓒ (f−g)(x)(f−g)(x) ⓓ (f−g)(−2).(f−g)(−2).

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Adding and Subtracting Polynomials

Practice Makes Perfect

Determine the Type of Polynomials

In the following exercises, determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.

ⓐ 47x5−17x2y3+y247x5−17x2y3+y2ⓑ 5c3+11c2−c−85c3+11c2−c−8ⓒ 59ab+13b59ab+13bⓓ 4ⓔ 4pq+174pq+17

ⓐ x2−y2x2−y2ⓑ −13c4−13c4ⓒ a2+2ab−7b2a2+2ab−7b2ⓓ 4x2y2−3xy+84x2y2−3xy+8ⓔ 19

ⓐ 8y−5x8y−5xⓑ y2−5yz−6z2y2−5yz−6z2ⓒ y3−8y2+2y−16y3−8y2+2y−16ⓓ 81ab4−24a2b2+3b81ab4−24a2b2+3bⓔ −18−18

ⓐ 11y211y2ⓑ −73−73ⓒ 6x2−3xy+4x−2y+y26x2−3xy+4x−2y+y2ⓓ 4y2+17z24y2+17z2ⓔ 5c3+11c2−c−85c3+11c2−c−8

ⓐ 5a2+12ab−7b25a2+12ab−7b2ⓑ 18xy2z18xy2zⓒ 5x+25x+2ⓓ y3−8y2+2y−16y3−8y2+2y−16ⓔ −24−24

Add and Subtract Polynomials

In the following exercises, add or subtract the monomials.

ⓐ 7x2+5x27x2+5x2ⓑ 4a−9a4a−9a

ⓐ 4y3+6y34y3+6y3ⓑ −y−5y−y−5y

ⓐ −12w+18w−12w+18wⓑ 7x2y−(−12x2y)7x2y−(−12x2y)

ⓐ −3m+9m−3m+9mⓑ 15yz2−(−8yz2)15yz2−(−8yz2)

7x 2 + 5 x 2 + 4a − 9 a 7x 2 + 5 x 2 + 4a − 9 a

4y 3 + 6 y 3 − y − 5 y 4y 3 + 6 y 3 − y − 5 y

−12 w + 18 w + 7 x 2 y − ( −12 x 2 y ) −12 w + 18 w + 7 x 2 y − ( −12 x 2 y )

−3 m + 9 m + 15 y z 2 − ( −8 y z 2 ) −3 m + 9 m + 15 y z 2 − ( −8 y z 2 )

ⓐ −5b−17b−5b−17bⓑ 3xy−(−8xy)+5xy3xy−(−8xy)+5xy

ⓐ −10x−35x−10x−35xⓑ 17mn2−(−9mn2)+3mn217mn2−(−9mn2)+3mn2

ⓐ 12a+5b−22a12a+5b−22aⓑ pq2−4p−3q2pq2−4p−3q2

ⓐ 14x−3y−13x14x−3y−13xⓑ a2b−4a−5ab2a2b−4a−5ab2

ⓐ 2a2+b2−6a22a2+b2−6a2ⓑ x2y−3x+7xy2x2y−3x+7xy2

ⓐ 5u2+4v2−6u25u2+4v2−6u2ⓑ 12a+8b12a+8b

ⓐ xy2−5x−5y2xy2−5x−5y2ⓑ 19y+5z19y+5z

12 a + 5 b − 22 a + p q 2 − 4 p − 3 q 2 12 a + 5 b − 22 a + p q 2 − 4 p − 3 q 2

14x − 3 y − 13 x + a 2 b − 4 a − 5 a b 2 14x − 3 y − 13 x + a 2 b − 4 a − 5 a b 2

2 a 2 + b 2 − 6 a 2 + x 2 y − 3 x + 7 x y 2 2 a 2 + b 2 − 6 a 2 + x 2 y − 3 x + 7 x y 2

5 u 2 + 4 v 2 − 6 u 2 + 12a + 8 b 5 u 2 + 4 v 2 − 6 u 2 + 12a + 8 b

x y 2 − 5 x − 5 y 2 + 19y + 5 z x y 2 − 5 x − 5 y 2 + 19y + 5 z

Add: 4a,−3b,−8a4a,−3b,−8a

Add:4x,3y,−3x4x,3y,−3x

Subtract 5x65x6 from −12x6−12x6

Subtract 2p42p4 from −7p4−7p4

In the following exercises, add the polynomials.

( 5 y 2 + 12 y + 4 ) + ( 6 y 2 − 8 y + 7 ) ( 5 y 2 + 12 y + 4 ) + ( 6 y 2 − 8 y + 7 )

( 4 y 2 + 10 y + 3 ) + ( 8 y 2 − 6 y + 5 ) ( 4 y 2 + 10 y + 3 ) + ( 8 y 2 − 6 y + 5 )

( x 2 + 6 x + 8 ) + ( −4 x 2 + 11 x − 9 ) ( x 2 + 6 x + 8 ) + ( −4 x 2 + 11 x − 9 )

( y 2 + 9 y + 4 ) + ( −2 y 2 − 5 y − 1 ) ( y 2 + 9 y + 4 ) + ( −2 y 2 − 5 y − 1 )

( 8 x 2 − 5 x + 2 ) + ( 3 x 2 + 3 ) ( 8 x 2 − 5 x + 2 ) + ( 3 x 2 + 3 )

( 7 x 2 − 9 x + 2 ) + ( 6 x 2 − 4 ) ( 7 x 2 − 9 x + 2 ) + ( 6 x 2 − 4 )

( 5 a 2 + 8 ) + ( a 2 − 4 a − 9 ) ( 5 a 2 + 8 ) + ( a 2 − 4 a − 9 )

( p 2 − 6 p − 18 ) + ( 2 p 2 + 11 ) ( p 2 − 6 p − 18 ) + ( 2 p 2 + 11 )

In the following exercises, subtract the polynomials.

( 4 m 2 − 6 m − 3 ) − ( 2 m 2 + m − 7 ) ( 4 m 2 − 6 m − 3 ) − ( 2 m 2 + m − 7 )

( 3 b 2 − 4 b + 1 ) − ( 5 b 2 − b − 2 ) ( 3 b 2 − 4 b + 1 ) − ( 5 b 2 − b − 2 )

( a 2 + 8 a + 5 ) − ( a 2 − 3 a + 2 ) ( a 2 + 8 a + 5 ) − ( a 2 − 3 a + 2 )

( b 2 − 7 b + 5 ) − ( b 2 − 2 b + 9 ) ( b 2 − 7 b + 5 ) − ( b 2 − 2 b + 9 )

( 12 s 2 − 15 s ) − ( s − 9 ) ( 12 s 2 − 15 s ) − ( s − 9 )

( 10 r 2 − 20 r ) − ( r − 8 ) ( 10 r 2 − 20 r ) − ( r − 8 )

In the following exercises, subtract the polynomials.

Subtract (9x2+2)(9x2+2) from (12x2−x+6)(12x2−x+6)

Subtract (5y2−y+12)(5y2−y+12) from (10y2−8y−20)(10y2−8y−20)

Subtract (7w2−4w+2)(7w2−4w+2) from (8w2−w+6)(8w2−w+6)

Subtract (5x2−x+12)(5x2−x+12) from (9x2−6x−20)(9x2−6x−20)

In the following exercises, find the difference of the polynomials.

Find the difference of (w2+w−42)(w2+w−42) and (w2−10w+24)(w2−10w+24)

Find the difference of (z2−3z−18)(z2−3z−18) and (z2+5z−20)(z2+5z−20)

In the following exercises, add the polynomials.

( 7 x 2 − 2 x y + 6 y 2 ) + ( 3 x 2 − 5 x y ) ( 7 x 2 − 2 x y + 6 y 2 ) + ( 3 x 2 − 5 x y )

( −5 x 2 − 4 x y − 3 y 2 ) + ( 2 x 2 − 7 x y ) ( −5 x 2 − 4 x y − 3 y 2 ) + ( 2 x 2 − 7 x y )

( 7 m 2 + m n − 8 n 2 ) + ( 3 m 2 + 2 m n ) ( 7 m 2 + m n − 8 n 2 ) + ( 3 m 2 + 2 m n )

( 2 r 2 − 3 r s − 2 s 2 ) + ( 5 r 2 − 3 r s ) ( 2 r 2 − 3 r s − 2 s 2 ) + ( 5 r 2 − 3 r s )

In the following exercises, add or subtract the polynomials.

( a 2 − b 2 ) − ( a 2 + 3 a b − 4 b 2 ) ( a 2 − b 2 ) − ( a 2 + 3 a b − 4 b 2 )

( m 2 + 2 n 2 ) − ( m 2 − 8 m n − n 2 ) ( m 2 + 2 n 2 ) − ( m 2 − 8 m n − n 2 )

( p 3 − 3 p 2 q ) + ( 2 p q 2 + 4 q 3 ) − ( 3 p 2 q + p q 2 ) ( p 3 − 3 p 2 q ) + ( 2 p q 2 + 4 q 3 ) − ( 3 p 2 q + p q 2 )

( a 3 − 2 a 2 b ) + ( a b 2 + b 3 ) − ( 3 a 2 b + 4 a b 2 ) ( a 3 − 2 a 2 b ) + ( a b 2 + b 3 ) − ( 3 a 2 b + 4 a b 2 )

( x 3 − x 2 y ) − ( 4 x y 2 − y 3 ) + ( 3 x 2 y − x y 2 ) ( x 3 − x 2 y ) − ( 4 x y 2 − y 3 ) + ( 3 x 2 y − x y 2 )

( x 3 − 2 x 2 y ) − ( x y 2 − 3 y 3 ) − ( x 2 y − 4 x y 2 ) ( x 3 − 2 x 2 y ) − ( x y 2 − 3 y 3 ) − ( x 2 y − 4 x y 2 )

Evaluate a Polynomial Function for a Given Value

In the following exercises, find the function values for each polynomial function.

In the following exercises, find the height for each polynomial function.

A painter drops a brush from a platform 75 feet high. The polynomial function h(t)=−16t2+75h(t)=−16t2+75 gives the height of the brush t seconds after it was dropped. Find the height after t=2t=2 seconds.

A girl drops a ball off a 200-foot cliff into the ocean. The polynomial h(t)=−16t2+200h(t)=−16t2+200 gives the height of the ball, in feet, t seconds after it is dropped. Find the height after t=3t=3 seconds.

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial function R(p)=−4p2+420p.R(p)=−4p2+420p. Find the revenue received when p=60p=60 dollars.

A manufacturer of the latest basketball shoes has found that the revenue received from selling the shoes at a cost of p dollars each is given by the polynomial R(p)=−4p2+420p.R(p)=−4p2+420p. Find the revenue received when p=90p=90 dollars.

The polynomial C(x)=6x2+90xC(x)=6x2+90x gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and height 6 feet. Find the cost of producing a box with x=4x=4 feet.

The polynomial C(x)=6x2+90xC(x)=6x2+90x gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and height 4 feet. Find the cost of producing a box with x=6x=6 feet.

Add and Subtract Polynomial Functions

f(x)=2x2−4x+1f(x)=2x2−4x+1 and g(x)=5x2+8x+3g(x)=5x2+8x+3

f(x)=4x2−7x+3f(x)=4x2−7x+3 and g(x)=4x2+2x−1g(x)=4x2+2x−1

f(x)=3x3−x2−2x+3f(x)=3x3−x2−2x+3 and g(x)=3x3−7xg(x)=3x3−7x

f(x)=5x3−x2+3x+4f(x)=5x3−x2+3x+4 and g(x)=8x3−1g(x)=8x3−1

Writing Exercises

Using your own words, explain the difference between a monomial, a binomial, and a trinomial.

Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of 5.

Ariana thinks the sum 6y2+5y46y2+5y4 is 11y6.11y6. What is wrong with her reasoning?

Is every trinomial a second degree polynomial? If not, give an example.

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…ਨਹੀਂ – ਮੈਨੂੰ ਇਹ ਸਮਝ ਨਹੀਂ ਆਉਂਦਾ! ਇਹ ਇੱਕ ਚੇਤਾਵਨੀ ਦਾ ਸੰਕੇਤ ਹੈ ਅਤੇ ਤੁਹਾਨੂੰ ਇਸਨੂੰ ਨਜ਼ਰਅੰਦਾਜ਼ ਨਹੀਂ ਕਰਨਾ ਚਾਹੀਦਾ। ਤੁਹਾਨੂੰ ਤੁਰੰਤ ਮਦਦ ਲੈਣੀ ਚਾਹੀਦੀ ਹੈ, ਨਹੀਂ ਤਾਂ ਤੁਸੀਂ ਜਲਦੀ ਹੀ ਹਾਵੀ ਹੋ ਜਾਵੋਗੇ। ਆਪਣੀ ਸਥਿਤੀ ਬਾਰੇ ਵਿਚਾਰ-ਵਟਾਂਦਰਾ ਕਰਨ ਲਈ ਜਿੰਨੀ ਜਲਦੀ ਹੋ ਸਕੇ ਆਪਣੇ ਇੰਸਟ੍ਰਕਟਰ ਨੂੰ ਮਿਲੋ। ਇਕੱਠੇ ਮਿਲ ਕੇ ਤੁਸੀਂ ਲੋੜੀਂਦੀ ਮਦਦ ਪ੍ਰਾਪਤ ਕਰਨ ਲਈ ਇੱਕ ਯੋਜਨਾ ਬਣਾ ਸਕਦੇ ਹੋ।