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College Algebra : Polynomials

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Example Questions

Example Question #1 : Polynomials

Multiply the following polynomials:

(4x^3+3x^2+9)(3x^2+4x-35)

Possible Answers:

12x^5+25x^4-128x^3-78x^2+36x-315

12x^5+25x^4-128x^3-132x^2+36x-315

12x^5+125x^4-28x^3-78x^2+36x-315

12x^5+25x^4-128x^3-78x^2

Correct answer:

12x^5+25x^4-128x^3-78x^2+36x-315

Explanation:

Multiply the following polynomials:

(4x^3+3x^2+9)(3x^2+4x-35)

When multiplying polynomials, we need to multiply each term in the first polynomial by each term in the second polynomial.

We can rewrite the expression as:

4x^3(3x^2+4x-35)+3x^2(3x^2+4x-35)+9(3x^2+4x-35)

Next, actually do the multiplication.

12x^5+16x^4-140x^3+9x^4+12x^3-105x^2+27x^2+36x-315

Finally, combine like terms to get the following:

12x^5+25x^4-128x^3-78x^2+36x-315

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Example Question #1 : Polynomials

Identify the power of the following polynomial

(x^6+2x-5^4+8)^4

Possible Answers:

Correct answer:

Explanation:

To find the power of a polynomial, you must find the term with the highest degree. Don't be fooled, you must distribute the power on the outside to the inside in order to find this term. Now, before you start worrying about foiling this 4 times, you can simply use the trick of distributing the outside power to the term on the inside with the largest power. Thus, our answer is:

6*4=24

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Example Question #1 : Polynomials

Multiply:

$(x^{2}+ 3x + 1 )(x^{2}- 3x + 1 )$

Possible Answers:

$x^{4}-9x^{2} + 6x - 1$

$x^{4}-11x^{2}+ 1$

$x^{4}-9x^{2} - 6x - 1$

$x^{4}+9x^{2} - 6x + 1$

$x^{4}-7x^{2}+ 1$

Correct answer:

$x^{4}-7x^{2}+ 1$

Explanation:

This product can be rewritten as the sum of two expressions multiplied by their difference:

$(x^{2}+ 3x + 1 )(x^{2}- 3x + 1 )$

$=(x^{2} + 1+ 3x )(x^{2}+ 1- 3x )$

$=[(x^{2} + 1)+ 3x][ (x^{2}+ 1)- 3x ]$

Apply the difference of squares pattern:

$= (x^{2}+ 1)^{2}- (3x)^{2}$

Apply the perfect square trinomial pattern on the left, and the Power of a Product Property on the right:

$=[ (x^{2})^{2}+2 \cdot x^{2} \cdot 1 + 1^{2}] - 3^{2}x^{2}$

$= (x^{4}+2 x^{2}+ 1 )-9x^{2}$

Collect like terms:

$= x^{4}+2 x^{2}-9x^{2}+ 1$

$= x^{4}-7x^{2}+ 1$

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Example Question #1 : Polynomials

Multiply

$(x^{2}- y- 1 )(x^{2}+ y + 1 )$ .

Possible Answers:

$x^{ 4}-y^{2} +1$

$x^{ 4}-y^{2} +2y-1$

$x^{ 4}-y^{2} - 2y-1$

$x^{ 4}-y^{2} -1$

$x^{ 4}+y^{2} +1$

Correct answer:

$x^{ 4}-y^{2} - 2y-1$

Explanation:

This product can be rewritten as the sum of two expressions multiplied by their difference:

$(x^{2}- y - 1 )(x^{2}+y + 1 )$

$=[x^{2}-( y + 1) ] [x^{2}+ (y + 1) ]$

Apply the difference of squares pattern:

$= (x^{2})^{2}-( y + 1)^{2}$

Now apply the Power of a Power Property on the left and the perfect square trinomial pattern on the right:

$= x^{ 2 \cdot 2}-( y^{2} + 2 \cdot y \cdot 1 + 1^{2})$

$= x^{ 4}-( y^{2} +2 y+ 1)$

$= x^{ 4}-y^{2} - 2y-1$

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College Algebra : Polynomials

Study concepts, example questions & explanations for College Algebra

All College Algebra Resources

Example Questions

Example Question #1 : Polynomials

Example Question #1 : Polynomials

Example Question #1 : Polynomials

Example Question #1 : Polynomials

All College Algebra Resources

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