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High School Math : Polynomials

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Example Questions

Example Question #31 : Intermediate Single Variable Algebra

Factor the following polynomial:

x^2-9-6xy+9y^2

Possible Answers:

(x-3y-3)(x-3y-3)

(x-3y+3)(x-3y-3)

(x-3y+3)(x+3y-3)

(x+3y+3)(x-3y-3)

(x+3y+3)(x+3y+3)

Correct answer:

(x-3y+3)(x-3y-3)

Explanation:

Begin by rearranging the terms to group together the quadratic:

x^2-9-6xy+9y^2

(x^2-6xy+9y^2)-9

Now, convert the quadratic into a square:

(x-3y)^2-9

Finally, distribute the :

(x-3y+3)(x-3y-3)

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Example Question #11 : Factoring Polynomials

Factor the following polynomial:

m^4n+m^2n-mn^4-mn^2

Possible Answers:

mn(m-n)(m^2+mn+n^2+1)

mn(m-n)(m^2-mn+n^2+1)

mn(m-n)(m^2+mn+n^2-1)

mn(m+n)(m^2+mn+n^2+1)

mn(m-n)(m^2+mn-n^2+1)

Correct answer:

mn(m-n)(m^2+mn+n^2+1)

Explanation:

Begin by extracting from the polynomial:

m^4n+m^2n-mn^4-mn^2

mn(m^3+m-n^3-n)

Now, rearrange to combine like terms:

mn(m^3-n^3+m-n)

Extract the like terms and factor the cubic:

mn[(m-n)(m^2+mn+n^2)+1(m-n)]

Simplify by combining like terms:

mn(m-n)(m^2+mn+n^2+1)

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Example Question #33 : Intermediate Single Variable Algebra

Factor the following polynomial:

3x^3+9x^2+9x+3

Possible Answers:

3(x-1)^2(x+1)

3(x-1)^3

3(x+1)^3

3(x+1)^2(x-1)

Correct answer:

3(x+1)^3

Explanation:

Begin by extracting from the polynomial:

3x^3+9x^2+9x+3

3(x^3+3x^2+3x+1)

Now, rearrange to combine like terms:

3(x^3+1+3x^2+3x)

Extract the like terms and factor the cubic:

3[(x+1)(x^2-x+1)+3x(x+1)]

Simplify by combining like terms:

3[(x+1)(x^2-x+1+3x)]

3(x+1)(x^2+2x+1)

3(x+1)(x+1)(x+1)

3(x+1)^3

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Example Question #1 : Write A Polynomial Function From Its Zeros

Consider the equation $14x^{4} +Ax^{3}+Bx^{2}+Cx+6 = 0$ .

According to the Rational Zeroes Theorem, if A,B,C are all integers, then, regardless of their values, which of the following cannot be a solution to the equation?

Possible Answers:

x=3

$x= \frac{3}{14}$

$x=\frac{1}{2}$

$x=\frac{10}{3}$

$x = \frac{6}{7}$

Correct answer:

$x=\frac{10}{3}$

Explanation:

By the Rational Zeroes Theorem, any rational solution must be a factor of the constant, 6, divided by the factor of the leading coefficient, 14.

Four of the answer choices have this characteristic:

$3 = 3 \div 1$

$\frac{1}{2} = 1\div 2$

$\frac{6}{7} = 6 \div 7$

$\frac{3}{14} =3 \div14$

$\frac{10}{3}$ is in lowest terms, and 3 is not a factor of 14. It is therefore the correct answer.

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High School Math : Polynomials

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #31 : Intermediate Single Variable Algebra

Example Question #11 : Factoring Polynomials

Example Question #33 : Intermediate Single Variable Algebra

Example Question #1 : Write A Polynomial Function From Its Zeros

All High School Math Resources

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