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Algebra 1 : Polynomials

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Example Questions

Example Question #84 : Expressions

Multiply: $(x -5) (x^{2} +5x + 25)$

Possible Answers:

$x^{3} -20x +125$

$x^{3} -125$

$x^{3} -20x -125$

$x^{3} +125$

$x^{3} -20x^{2} -125$

Correct answer:

$x^{3} -125$

Explanation:

$(x -5) (x^{2} +5x + 16)$

$= x (x^{2} +5x + 25)-5 (x^{2} +5x + 25)$

$= x \cdot x^{2} +x \cdot 5x + x \cdot 25-5 \cdot x^{2} -5 \cdot 5x -5 \cdot 25$

$= x^{3} +5x^{2} +25x-5 x^{2} -25x -125$

$= x^{3} +5x^{2} -5 x^{2} +25x -25x -125$

$= x^{3} -125$

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Example Question #22 : Intermediate Single Variable Algebra

Write as a polynomial in standard form: $(3x - 2y)^{3}$

Possible Answers:

$27x^{3} - 18 x^{2} y + 12 xy^{2} - 8y^{3}$

$27x^{3} - 54 x^{2} y + 36 xy^{2} - 8y^{3}$

$27x^{3} - 8y^{3}$

$27x^{3} + 18 x^{2} y - 12 xy^{2} - 8y^{3}$

$27x^{3} + 54 x^{2} y - 36 xy^{2} - 8y^{3}$

Correct answer:

$27x^{3} - 54 x^{2} y + 36 xy^{2} - 8y^{3}$

Explanation:

$(A -B)^{3} = A^{3} - 3A^{2} B + 3AB^{2} - B^{3}$

Replace: A = 3x, B = 2y :

$(3x -2y)^{3}$

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

$= 3^{3}x^{3} - 3 \cdot 3^{2} \cdot 2 \cdot x^{2} y + 3 \cdot 3 \cdot 2^{2} \cdot xy^{2} - 2^{3}y^{3}$

$= 27x^{3} - 54 x^{2} y + 36 xy^{2} - 8y^{3}$

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Example Question #2 : Multiplication And Division

Multiply: (x - 4)(x+ 4) (x -7)

Possible Answers:

$x^{3} - 7x^{2} - 16x + 112$

$x^{3} + 112$

$x^{3} -23x + 112$

$x^{3} -23x^{2} + 112$

$x^{3} + 7x^{2} - 16x - 112$

Correct answer:

$x^{3} - 7x^{2} - 16x + 112$

Explanation:

The first two factors are the product of the sum and the difference of the same two terms, so we can use the difference of squares:

(x - 4)(x+ 4) (x -7)

$= (x^{2} - 4^{2}) (x -7)$

$= (x^{2} - 16) (x -7)$

Now use the FOIL method:

$= x^{2}\cdot x - x^{2} \cdot 7 - 16 \cdot x + 16 \cdot7$

$= x^{3} - 7x^{2} - 16x + 112$

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Example Question #14 : Simplifying Exponents

Simplify the expression.

y^2z^3y^4z^5y^6

Possible Answers:

$y^{12}z^{10}$

$y^{15}z^3$

$y^{12}z^8$

$y^{10}z^8$

$y^9z^{11}$

Correct answer:

$y^{12}z^8$

Explanation:

y^2z^3y^4z^5y^6

Rearrange the expression so that the $\small y$ and $\small z$ variables of different powers are right next to each other.

y^2y^4y^6z^3z^5

When multiplying the same variable with different exponents, it is the same as adding the exponents: $a^ba^c=a^{b+c}$ . Taking advantage of this rule, the problem can be rewritten.

$y^{2+4+6}z^{3+5}$

$y^{6+6}z^8$

$y^{12}z^8$

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Example Question #41 : Polynomial Operations

Rewrite as a single radical expression, assuming is positive:

$\sqrt[4]{x} \cdot \sqrt[3]{x}$

Possible Answers:

$\dpi{150} x\; \sqrt[7]{x ^{5}}$

$\dpi{150} \sqrt[12]{x}$

$\dpi{150} \sqrt[12]{x ^{7}}$

$\dpi{150} \sqrt[7]{x}$

$\dpi{150} \sqrt[7]{x ^{12}}$

Correct answer:

$\dpi{150} \sqrt[12]{x ^{7}}$

Explanation:

Rewrite each radical as an exponential expression, apply the product of powers property, and rewrite the product as a radical:

$\sqrt[4]{x} \cdot \sqrt[3]{x}$

$=x^{ \frac{1}{4}} \cdot x^{ \frac{1}{3}}$

$=x^{ \frac{1}{4}+ \frac{1}{3}}$

$=x^{ \frac{3}{12}+ \frac{4}{12}}$

$=x^{ \frac{7}{12}}$

$=\sqrt[12]{x ^{7}}$

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Example Question #41 : Polynomial Operations

Evaluate: (x-2-x^2)(4-x+3x^2)

Possible Answers:

-3x^4+4x^3+11x^2-6x-8

-3x^4+4x^3-11x^2+6x-8

-3x^4+4x^3-9x^2-6x-8

-3x^4+3x^3+11x^2+6x-8

-3x^4+4x^3-9x^2+6x-8

Correct answer:

-3x^4+4x^3-11x^2+6x-8

Explanation:

This set is out of order, and it may be best to reorganize the terms. To solve, this is very similar to the FOIL method.

(x-2-x^2)(4-x+3x^2)= (-x^2+x-2)(3x^2-x+4)

Follow the procedure to distribute each term.

(a+b+c)(e+f+g)

= (ae+af+ag)+(be+bf+bg)+(ce+cf+cg)

=a(e+f+g)+b(e+f+g)+c(e+f+g)

Follow suit to solve the problem.

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

=(-x^2)(3x^2-x+4)+(x)(3x^2-x+4)+(-2)(3x^2-x+4)

=(-3x^4+x^3-4x^2)+(3x^3-x^2+4x)-6x^2+2x-8

Combine like terms.

The correct answer is: -3x^4+4x^3-11x^2+6x-8

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Example Question #14 : How To Multiply Polynomials

Multiply and simplify the expression
$\frac{3a^2b^4c}{4ab^2c^3} X \frac{12a^5b^2c}{3a^2bc^2}$ .

Possible Answers:

$\frac{4b^2c}{2a^2}$

$\frac{36a^3c^2}{b^2}$

$\frac{3a^2b^2}{c^5}$

3a^3b^2c

$\frac{3a^4b^3}{c^3}$

Correct answer:

$\frac{3a^4b^3}{c^3}$

Explanation:

The question is asking for the simplified version of this expression:

$\frac{3a^2b^4c}{4ab^2c^3} X \frac{12a^5b^2c}{3a^2bc^2}$

First, combine like terms in the numerator and denominator according to multiplication and exponent rules. When you are multiplying like bases together you add their exponents.

$\frac{3*12*a^2*a^5*b^4*b^2*c*c}{4*3*a*a^2*b^2*b*c^3*c^2}$

When you divide like bases you subtract the bottom exponent from the top exponent.

$\frac{36a^7b^6c^2}{12a^3b^3c^5} = (36/12)a^{7-3}b^{6-3}c^{2-5}$

Finally, simplify the expression by cancelling out terms using GCFs.

$3a^{4}b^{3}c^{-3}$

$\frac{3a^4b^3}{c^3}$

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Example Question #15 : How To Multiply Polynomials

Multiply the polynomials and simplify: $(4x^3+2x^2-8)(x-\frac{1}{4})$

Possible Answers:

$4x^4+x^3-\frac{1}{2}x^2-8x-2$

$4x^4+3x^3-\frac{1}{2}x^2-8x+2$

$4x^4-x^3-\frac{1}{2}x^2-8x-2$

$4x^4+x^3-\frac{1}{2}x^2-8x+2$

$4x^4+x^3-\frac{1}{4}x^2-8x+2$

Correct answer:

$4x^4+x^3-\frac{1}{2}x^2-8x+2$

Explanation:

The key to multiplying polynomials is to keep track of signs and be absolutely sure that you multiply every term in one polynomial (choose the first one) by every term in the other polynomial. For example if polynomial A has 3 terms and polynomial B has two, then then start with the first term in polynomial A and multiply it by both terms in polynomial B. Then choose the second term in polynomial A and repeat. Continue like so until you have gone through all iterations.

First term:

$({\color{Red} 4x^3}+2x^2-8)({\color{Red} x-\frac{1}{4}})$

4x^4-x^3

Second term:

$(4x^3+{\color{Green} 2x^2}-8)({\color{Green} x-\frac{1}{4}})$

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

Third term:

$(4x^3+2x^2-{\color{DarkBlue} 8})({\color{DarkBlue} x-\frac{1}{4}})$

$4x^4-x^3+2x^3-\frac{1}{2}x^2-8x+2$

Now combine terms where able for the final answer:

$4x^4+x^3-\frac{1}{2}x^2-8x+2$

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Example Question #16 : How To Multiply Polynomials

Multiply: (x^2-x-3)(x^2+x+3)

Possible Answers:

x^4-x^2-6x-9

x^4+3x^3-x^2+6x-9

x^4+3x^3-x^2-6x-9

x^4-2x^3+x^2-9

x^4-2x^3-5x^2+6x+9

Correct answer:

x^4-x^2-6x-9

Explanation:

Multiply each term of the first polynomial with all the terms of the second polynomial. Follow the signs of the first polynomial.

x^2(x^2+x+3) =x^4+x^3+3x^2

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

-3(x^2+x+3)= -3x^2-3x-9

Add and combine like terms.

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

The answer is: x^4-x^2-6x-9

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Example Question #1 : How To Subtract Polynomials

Subtract the polynomials below:

(14a^2 + 2a - 5) - (10a^2 - 3a + 8)

Possible Answers:

4a^2 - a + 3

4a^2 + 5a - 13

24a^2 - a + 13

4a^2 + 5a + 3

4a^2 + 5a - 3

Correct answer:

4a^2 + 5a - 13

Explanation:

The first step is to get everything out of parentheses to combine like terms. Since the polynomials are being subtracted, the sign of everything in the second polynomial will be flipped. You can think of this as a being distributed across the polynomial:

(14a^2 + 2a - 5) - (10a^2 - 3a + 8)

=(14a^2 + 2a - 5) + -1(10a^2 - 3a + 8)

=14a^2 + 2a - 5 - 10a^2 + 3a - 8

Now combine like terms:

14a^2 + 2a - 5 - 10a^2 + 3a - 8

=4a^2 + 2a - 5 + 3a - 8

=4a^2 + 5a - 5 - 8

=4a^2 + 5a - 13

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Algebra 1 : Polynomials

Study concepts, example questions & explanations for Algebra 1

All Algebra 1 Resources

Example Questions

Example Question #84 : Expressions

Example Question #22 : Intermediate Single Variable Algebra

Example Question #2 : Multiplication And Division

Example Question #14 : Simplifying Exponents

Example Question #41 : Polynomial Operations

Example Question #41 : Polynomial Operations

Example Question #14 : How To Multiply Polynomials

Example Question #15 : How To Multiply Polynomials

Example Question #16 : How To Multiply Polynomials

Example Question #1 : How To Subtract Polynomials

All Algebra 1 Resources

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