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Algebra 1 : Polynomials

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Example Questions

Example Question #1 : How To Multiply Polynomials

Multiply:

$( 3x^{2}-2x+5 )\cdot ( - x^{3}+2x^{2}-5x-10 )$

Possible Answers:

$-3x^{4}+8x^{3}-24x^{2}-10x-55$

$-3x^{5}-8x^{4}+24x^{3}+10x^{2}+5x+50$

$3x^{4}+8x^{3}-24x^{2}-15x-50$

$-3x^{5}+8x^{4}-24x^{3}-10x^{2}-5x-50$

$-3x^{5}+4x^{4}-16x^{3}-4-30x^{2}-45x+50$

Correct answer:

$-3x^{5}+8x^{4}-24x^{3}-10x^{2}-5x-50$

Explanation:

First multiply each of the terms inside the second parentheses by $3x^{2}$ :

$-3x^{5}+6x^{4}-15x^{3}-30x^{2}$

Next multiply each of the terms inside the second parentheses by -2x :

$2x^{4}-4x^{3}+10x^{2}+20x$

Finally multiply each of the terms inside the second parentheses by 5:

$-5x^{3}+10x^{2}-25x-50$

By combining all like terms we get the correct answer:

$-3x^{5}+8x^{4}-24x^{3}-10x^{2}-5x-50$

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Example Question #91 : Variables

Expand:

(x + 3)(x^2 - 2x + 5)

Possible Answers:

x^3 + x^2 + x + 15

x^3 - x^2 - x + 15

x^3 + 5x^2 - x + 15

x^3 + x^2 - x - 15

x^3 + x^2 - x + 15

Correct answer:

x^3 + x^2 - x + 15

Explanation:

The first step is to distribute the first term in the binomial across the trinomial:

x(x^2 - 2x + 5) = x^3 - 2x^2 + 5x

Then multiply the second term in the binomial across the same trinomial:

3(x^2 - 2x + 5) = 3x^2 - 6x + 15

You can then combine like terms to reach a simplified answer:

(x+3)(x^2 - 2x + 5) = x^3 - 2x^2 + 5x + 3x^2 - 6x + 15

=x^3 - 2x^2 + 5x + 3x^2 - 6x + 15

=x^3 + x^2 - x + 15

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Example Question #91 : Polynomials

Multiply the expression: (x^2 + y^2) (2x - 4)

Possible Answers:

x^3-4 x^2+2 x y^2- y^2

2 x^3-4 x^2+2 x y^2-4 y^2

2 x^3-4 x^2+2 x^2 y^2-4 y

2 x^2-4 x+2 x y^2-4 y^2

2 x^3-4 x^2 + y^2

Correct answer:

2 x^3-4 x^2+2 x y^2-4 y^2

Explanation:

When we FOIL this expression, we start by multiplying the first term of the first binomial with the first term of the second binomial. Then we multiply the two "outside" terms (in this case those are x^2 and 4). Next we multiply the "inside" terms (in this case the two inside terms are y^2 and ) and finally the last terms.

When we combine terms in this way, it is important to make sure we are paying attention to the exponents! We want to add the exponents when we multiply terms. So, when multiplying $(x^2)\cdot (x)$ , we want to add the 2 with 1 (the exponent of an un-powered variable is always 1!) and we get 3! So we know the resulting power would be 3, like this: x^3

Using that skill and the FOIL technique, we end up with the answer, 2 x^3-4 x^2+2 x y^2-4 y^2

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Example Question #5 : How To Multiply Polynomials

Expand: (2x + 6) (5x^2 - 5)

Possible Answers:

10 x^2+30 x-30

x^3+30 x^2- x-30

10 x^3+30 x^2 +10 x+30

10 x^3+30 x^2-10 x-30

10 x^3+30 x^2-10 x

Correct answer:

10 x^3+30 x^2-10 x-30

Explanation:

When we FOIL this expression, we start by multiplying the first term of the first binomial with the first term of the second binomial. Then we multiply the two "outside" terms (in this case those are the and ). Next we multiply the "inside" terms (in this case the two inside terms are and 5x^2 ) and finally the last terms.

When we combine terms in this way, it is important to make sure we are paying attention to the exponents! When we multiply like terms with exponents, we can add the exponents together! In this case, we have an (which since it has no power, means it is to the power of 1) and an x^2 . We can add these exponents together when we go through our FOIL process and get . So we know the resulting power would be , like this: x^3

Using that skill and the FOIL technique, we end up with the answer:

10 x^3+30 x^2-10 x-30

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Example Question #1 : Simplifying Polynomials

Multiply:

$\left (x^{2} + 2x + 7 \right ) \left (x^{2} + x - 6 \right )$

Possible Answers:

$x^{4} + 3 x^{3} -x^{2} - 5x - 42$

$x^{4} + 3 x^{3} -15x^{2} - 19x - 42$

$x^{4} + 3 x^{3} + 19x^{2} - 5x - 42$

$x^{4} + 3 x^{3} +3x^{2} - 5x - 42$

$x^{4} + 3 x^{3} -x^{2} - 19x - 42$

Correct answer:

$x^{4} + 3 x^{3} +3x^{2} - 5x - 42$

Explanation:

$\left (x^{2} + 2x + 7 \right ) \left (x^{2} + x - 6 \right )$

$= x^{2}\left (x^{2} + x - 6 \right ) + 2x \left (x^{2} + x - 6 \right ) + 7 \left (x^{2} + x - 6 \right )$

$= x^{2} \cdot x^{2} + x^{2} \cdot x - x^{2} \cdot 6 + 2x \cdot x^{2} + 2x \cdot x - 2x \cdot 6 + 7 \cdot x^{2} + 7 \cdot x - 7 \cdot 6$

$= x^{4} + x^{3} - 6x^{2} + 2x^{3} + 2x^{2} - 12x + 7x^{2} +7x - 42$

$= x^{4} + x^{3} + 2x^{3} - 6x^{2} + 2x^{2} + 7x^{2} - 12x +7x - 42$

$= x^{4} + 3 x^{3} +3x^{2} - 5x - 42$

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Example Question #1 : Simplifying Polynomials

Simplify the expression.

$\small (3x+2)-(x+4)(x+1)$

Possible Answers:

$\small -x^2-2x-2$

$\small -x^2+8x+6$

$\small -x^2-8x-6$

$\small x^2+8x+6$

Correct answer:

$\small -x^2-2x-2$

Explanation:

$\small (3x+2)-(x+4)(x+1)$

Use FOIL to expand the monomials.

(x+4)(x+1)

x^2+x+4x+4=x^2+5x+4

Return this expansion to the original expression.

$\small (3x+2)-(x^2+5x+4)$

Distribute negative sign.

$\small (3x+2)-x^2-5x-4$

Combine like terms.

$\small -x^2+3x-5x+2-4$

$\small -x^2-2x-2$

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Example Question #1 : How To Multiply Polynomials

Simplify the expression.

$x^{5}*x^{16}$

Possible Answers:

The expression cannot be simplified further.

$x^{80}$

$2x^{16}$

$x^{21}$

$x^{11}$

Correct answer:

$x^{21}$

Explanation:

When multiplying exponential components, you must add the powers of each term together.

5+16=21

$x^{5}*x^{16}=x^{21}$

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Example Question #2 : How To Multiply Polynomials

$x^{5}y^{17}*x^{2}y^{2}=?$

Possible Answers:

None of the other answers are correct.

$x^{7}y^{19}$

$x^{2}y^{8}$

$x^{10}y^{34}$

$2x^{2}x^{7}y^{19}$

Correct answer:

$x^{7}y^{19}$

Explanation:

When multiplying polynomials, add the powers of each like-termed variable together.

For x: 5 + 2 = 7

For y: 17 + 2 = 19

Therefore the answer is $x^{7}y^{19}$ .

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Example Question #2 : Simplifying And Expanding Quadratics

Multiply: $(x^{2} + 3) (4x - 1)$

Possible Answers:

$4x^{3} +x^{2} -12x - 3$

$4x^{3} + 3$

$4x^{3} -x^{2} +12x - 3$

$4x^{3} - 3$

$4x^{3} -x^{2} -12x + 3$

Correct answer:

$4x^{3} -x^{2} +12x - 3$

Explanation:

$(x^{2} + 3) (4x - 1)$

$= x^{2} (4x - 1) + 3(4x - 1)$

$= x^{2} \cdot 4x - x^{2} \cdot 1 + 3 \cdot 4x - 3 \cdot 1$

$= 4x^{3} - x^{2} + 12x - 3$

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Example Question #2 : Simplifying And Expanding Quadratics

Multiply: $(x + 4) (x^{2} -4x + 16)$

Possible Answers:

$x^{3}- 12x - 64$

$x^{3}- 20x + 64$

$x^{3}- 12x + 64$

$x^{3} - 64$

$x^{3} + 64$

Correct answer:

$x^{3} + 64$

Explanation:

$(x + 4) (x^{2} -4x + 16)$

$= x (x^{2} -4x + 16)+ 4 (x^{2} -4x + 16)$

$= x \cdot x^{2} -x \cdot 4x + x \cdot 16+ 4 \cdot x^{2} -4 \cdot 4x + 4 \cdot 16$

$= x^{3} -4x^{2} + 16x+ 4 x^{2} -16x + 64$

$= x^{3} -4x^{2} + 4 x^{2} + 16x -16x + 64$

$= x^{3} + 64$

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Algebra 1 : Polynomials

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #1 : How To Multiply Polynomials

Example Question #91 : Variables

Example Question #91 : Polynomials

Example Question #5 : How To Multiply Polynomials

Example Question #1 : Simplifying Polynomials

Example Question #1 : Simplifying Polynomials

Example Question #1 : How To Multiply Polynomials

Example Question #2 : How To Multiply Polynomials

Example Question #2 : Simplifying And Expanding Quadratics

Example Question #2 : Simplifying And Expanding Quadratics

All Algebra 1 Resources

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