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Common Core: High School - Algebra : Arithmetic with Polynomials & Rational Expressions

Study concepts, example questions & explanations for Common Core: High School - Algebra

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All Common Core: High School - Algebra Resources

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Example Questions

Example Question #1 : Arithmetic With Polynomials & Rational Expressions

Given and find A-B .

$\\A=x^2+2 \\B=4x^2-1$

Possible Answers:

-3x^2-3

3x^2-3

-3x^2+3

3x^2+3

-5x^2+3

Correct answer:

-3x^2+3

Explanation:

To find the difference of two polynomials first set up the operation and identify the like terms.

$\\A=x^2+2 \\B=4x^2-1$

A-B= (x^2+2)-(4x^2-1)

The like terms in these polynomials are the squared variable and the constant terms.

Remember to distribute the negative sign through to all terms in the second polynomial.

$A-B={\color{Red} x^2}{\color{Blue} +2}{\color{Red} -4x^2}{\color{Blue} --1}$

$\\x^2-4x^2=-3x^2 \\2-(-1)=3$

Therefore, the sum of these polynomials is,

-3x^2+3

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Example Question #2 : Arithmetic With Polynomials & Rational Expressions

Given and find A+B .

$\\A=x^2+2 \\B=4x^2-1$

Possible Answers:

3x^2-1

3x^2+1

5x^2+1

5x^2+3

5x^2-1

Correct answer:

5x^2+1

Explanation:

To find the sum of two polynomials first set up the operation and identify the like terms.

$\\A=x^2+2 \\B=4x^2-1$

A+B= (x^2+2)+(4x^2-1)

The like terms in these polynomials are the squared variable and the constant terms.

$A+B={\color{Red} x^2}{\color{Blue} +2}{\color{Red} +4x^2}{\color{Blue} -1}$

$\\x^2+4x^2=5x^2 \\2+(-1)=1$

Therefore, the sum of these polynomials is,

5x^2+1

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Example Question #3 : Working With Complex Polynomials

Given and find B-A .

$\\A=x^2+2 \\B=4x^2-1$

Possible Answers:

-3x^2-3

-3x^2+3

5x^2-3

3x^2+3

3x^2-3

Correct answer:

3x^2-3

Explanation:

To find the difference of two polynomials first set up the operation and identify the like terms.

$\\A=x^2+2 \\B=4x^2-1$

B-A= (4x^2-1)-(x^2+2)

The like terms in these polynomials are the squared variable and the constant terms.

Remember to distribute the negative sign through to all terms in the second polynomial.

$B-A={\color{Red} 4x^2}{\color{Blue} -1}{\color{Red} -x^2}{\color{Blue} -2}$

$\\4x^2-x^2=3x^2 \\-1-2=-3$

Therefore, the sum of these polynomials is,

3x^2-3

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Example Question #4 : Working With Complex Polynomials

Given and find A-B .

$\\A=3x^3+x+2 \\B=4x^3-2x-1$

Possible Answers:

x^3+3x+3

-x^3-3x+3

-x^3+3x+3

x^3+3x-3

-x^3-3x-3

Correct answer:

-x^3+3x+3

Explanation:

To find the difference of two polynomials first set up the operation and identify the like terms.

$\\A=3x^3+x+2 \\B=4x^3-2x-1$

A-B= (3x^3+x+2)-(4x^3-2x-1)

The like terms in these polynomials are the squared variable, the single variable, and the constant terms.

Remember to distribute the negative sign to all terms within the parentheses.

$A-B={\color{Red} 3x^3}{\color{Blue} +x}+2{\color{Red} -4x^3}{\color{Blue} +2x}+1$

$\\3x^3-4x^3=-x^3 \\x+2x=3x \\2+1=3$

Therefore, the sum of these polynomials is,

-x^3+3x+3

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Example Question #5 : Working With Complex Polynomials

Given and find A+B .

$\\A=2x^2+2 \\B=4x^2-2$

Possible Answers:

6x^2-4

-6x^2

6x^2+2

6x^2

6x^2+4

Correct answer:

6x^2

Explanation:

To find the sum of two polynomials first set up the operation and identify the like terms.

$\\A=2x^2+2 \\B=4x^2-2$

A+B= (2x^2+2)+(4x^2-2)

The like terms in these polynomials are the squared variable and the constant terms.

$A+B={\color{Red} 2x^2}{\color{Blue} +2}{\color{Red} +4x^2}{\color{Blue} -2}$

$\\2x^2+4x^2=6x^2 \\2+(-2)=0$

Therefore, the sum of these polynomials is,

6x^2

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Example Question #6 : Working With Complex Polynomials

Given and find A-B .

$\\A=2x^2+2 \\B=4x^2-2$

Possible Answers:

2x^2+4

-2x^2+4

2x^2-4

-2x^2+2

-2x^2-4

Correct answer:

-2x^2+4

Explanation:

To find the difference of two polynomials first set up the operation and identify the like terms.

$\\A=2x^2+2 \\B=4x^2-2$

A-B= (2x^2+2)-(4x^2-2)

The like terms in these polynomials are the squared variable and the constant terms.

Remember to distribute the negative sign to all terms in the second polynomial.

$A-B={\color{Red} 2x^2}{\color{Blue} +2}{\color{Red} -4x^2}{\color{Blue} +2}$

$\\2x^2-4x^2=-2x^2 \\2+2=4$

Therefore, the sum of these polynomials is,

-2x^2+4

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Example Question #2 : Polynomials And Quadratics

Given and find B-A .

$\\A=2x^2+2 \\B=4x^2-2$

Possible Answers:

2x^2

2x^2-4

-2x^2+4

2x^2+4

-2x^2-4

Correct answer:

2x^2-4

Explanation:

To find the difference of two polynomials first set up the operation and identify the like terms.

$\\A=2x^2+2 \\B=4x^2-2$

B-A= (4x^2-2)-(2x^2+2)

The like terms in these polynomials are the squared variable and the constant terms.

Remember to distribute the negative sign to all terms in the second polynomial.

$B-A= {\color{Red} 4x^2}-2{\color{Red} -2x^2}-2$

$\\4x^2-2x^2=2x^2 \\-2-2=-4$

Therefore, the sum of these polynomials is,

2x^2-4

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Example Question #7 : Working With Complex Polynomials

Given and find A+B .

$\\A=x^2 \\B=4x^2-1$

Possible Answers:

-5x^2-1

-5x^2+1

5x^2

5x^2+1

5x^2-1

Correct answer:

5x^2-1

Explanation:

To find the sum of two polynomials first set up the operation and identify the like terms.

$\\A=x^2 \\B=4x^2-1$

A+B= (x^2)+(4x^2-1)

The like terms in these polynomials are the squared variable.

$A+B={\color{Red} x^2}{\color{Red} +4x^2}-1$

$\\x^2+4x^2=5x^2 \\0+(-1)=-1$

Therefore, the sum of these polynomials is,

5x^2-1

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Example Question #8 : Working With Complex Polynomials

Given and find .

$\\A=x^2+1 \\B=4x^2-1$

Possible Answers:

4x^4-3x^2-1

4x^4+3x^2+1

-4x^4-3x^2-1

4x^4+3x^2-1

-4x^4+3x^2-1

Correct answer:

4x^4+3x^2-1

Explanation:

To find the product of two polynomials first set up the operation.

$\\A=x^2+1 \\B=4x^2-1$

AB=(x^2+1)(4x^2-1)

Now, multiply each term from the first polynomial with each term in the second polynomial.

Remember the rules of exponents. When like base variables are multiplied together their exponents are added together.

$\\x^2\cdot 4x^2=4x^{2+2}=4x^4 \\x^2\cdot -1=-x^2 \\1\cdot 4x^2=4x^2 \\1\cdot -1=-1$

Therefore, the product of these polynomials is,

4x^4-x^2+4x^2-1

Combine like terms to arrive at the final answer.

4x^4+3x^2-1

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Example Question #9 : Working With Complex Polynomials

If $A=3x^{3}+x+2$ , and $B=4x^{3}-2x-1$ , what is the value of A+B?

Possible Answers:

7x^3-x+1

7x^3-x-1

7x^3+x-1

7x^3+x+1

-7x^3-x+1

Correct answer:

7x^3-x+1

Explanation:

In order to find the sum of two polynomials, we must first set up the operation and identify the like terms.

$\\A=3x^3+x+2 \\B=4x^3-2x-1$

A+B= (3x^3+x+2)+(4x^3-2x-1)

The like terms in these polynomials are the squared variable, the single variable, and the constant terms.

A+B= 3x^3+x+2+4x^3-2x-1

$\\3x^3+4x^3=7x^3 \\x-2x=-x \\2-1=1$

The sum of these polynomials is equal to the following expression:

7x^3-x+1

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Common Core: High School - Algebra : Arithmetic with Polynomials & Rational Expressions

Study concepts, example questions & explanations for Common Core: High School - Algebra

All Common Core: High School - Algebra Resources

Example Questions

Example Question #1 : Arithmetic With Polynomials & Rational Expressions

Example Question #2 : Arithmetic With Polynomials & Rational Expressions

Example Question #3 : Working With Complex Polynomials

Example Question #4 : Working With Complex Polynomials

Example Question #5 : Working With Complex Polynomials

Example Question #6 : Working With Complex Polynomials

Example Question #2 : Polynomials And Quadratics

Example Question #7 : Working With Complex Polynomials

Example Question #8 : Working With Complex Polynomials

Example Question #9 : Working With Complex Polynomials

All Common Core: High School - Algebra Resources

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