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ACT Math : How to subtract polynomials

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Example Questions

Example Question #1 : How To Subtract Polynomials

Simplify:

$(4x^{2}-6x^{5}-2)-(x^{5}+3x^{3}-4x^{2}-2)$

Possible Answers:

$-7x^{5}-3x^{3}+ 8x^{2}-4$

${ -7x^{5}}-3x^{3}+ 8x^{2}}$

$-7x^{5}+3x^{3}-8x^{2}$

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

$-7x^{5}+3x^{3}-4$

Correct answer:

${ -7x^{5}}-3x^{3}+ 8x^{2}}$

Explanation:

When subtracting polynomials, it's helpful to remember that the "minus sign" gets distributed. It's as if the two polynomials are being added and a -1 is in front of the second polynomial.

$(4x^{2}-6x^{5}-2){\color{Blue} +}{\color{Red} (-1)}(x^{5}+3x^{3}-4x^{2}-2)$

This -1 will get multiplied to all the terms in the second polynomial that is being subtracted from the first, so it becomes $({\color{red} -}x^{5}{\color{Red} -}3x^{3}{\color{Red} +}4x^{2}{\color{Red} +}2)$ . You may note that by multiplying by -1, every term in the polynomial switches its original sign. The problem then becomes:

$(4x^{2}-6x^{5}-2)+(-x^{5}-3x^{3}+4x^{2}+2)$

From here, in order to simplify, because there's no equal sign, it can be deduced that we are not working toward a solution for x. The original problem is presented as an expression so an expression as an answer will be expected. In order to work towards a final simplified expression, like terms must be collected. This will provide the final answer.

${\color{Magenta} 4x^{2}}-{\color{DarkRed} 6x^{5}}{\color{Teal} -2}+{\color{DarkRed} -x^{5}}-3x^{3}+{\color{Magenta} 4x^{2}}{\color{Teal} +2}$

${\color{DarkRed} -7x^{5}}-3x^{3}+{\color{Magenta} 8x^{2}}$

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Example Question #1 : How To Subtract Polynomials

Simplify:

3x^2 + 5x +10 - (10x^2-15x-3)

Possible Answers:

13x^2 +13

-7x^2 - 10x +7

-7x^2 + 20x +13

3x^2 +5x +7

-17x^2+7

Correct answer:

-7x^2 + 20x +13

Explanation:

Begin by distributing the subtraction of the second term in this question:

3x^2 + 5x +10 -10x^2+15x+3

Now, you merely need to combine like terms:

3x^2 -10x^2 + 5x +15x +10 +3

-7x^2 + 20x +13

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Example Question #12 : Polynomials

If $f(x) = 2x^{2} - 3$ , then what does f(x + a) equal?

Possible Answers:

$2x^{2} - 4ax + 2a^{2} - 3$

$4x^{2} + 8ax + 4a^{2} - 3$

$2x^{2} + 4ax + 2a^{2} - 3$

$4x^{2} + 4ax + 4a^{2} - 3$

$2x^{2} + 2ax + 2a^{2} - 3$

Correct answer:

$2x^{2} + 4ax + 2a^{2} - 3$

Explanation:

To solve this equation, we substitute (x+a) in for every instance of seen in the original equation $f(x)=2x^{2}-3$ .

Therefore the new equation would read

$f(x)=2x^{2}-3\rightarrow f(x+a) = 2(x + a)^{2} -3$

Now we must square the expression x+a . To do this, you must multiply the expression by itself. Therefore:

$(x+a)^{2}=(x+a)\cdot(x+a)$

$(x+a)\cdot(x+a)=(x\cdot x)+(x\cdot a)+(x\cdot a)+(a\cdot a)$

$(x\cdot x)+(x\cdot a)+(x\cdot a)+(a\cdot a)=x^{2}+2ax+a^{2}$

We must now plug in our new value for $(x+a)^{2}$ into our original equation in place of $x^{2}$ .

$f(x)=2(x^{2}+2ax+a^{2})-3$

Now we must distribute the into $(x^{2}+2ax+a^{2})$ . To do this, you multiply each expression within the parenthesis by :

$2(x^{2}+2ax+a^{2})=2x^{2}+4ax+2a^{2}$

Therefore, our answer is $f(x)=2x^{2}+4ax+2a^{2}-3$ .

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Example Question #12 : Polynomials

The expression a[(b-c) +d] is equivalent to which of the following?

Possible Answers:

ab+ac-ad

ab-ac+ad

b-c+ad

ab-ac-ad

ab+ac+ad

Correct answer:

ab-ac+ad

Explanation:

To answer this question, we must distribute the to the rest of the variables , , and that are within the brackets.

To distribute a variable or number, you multiply that value with every other value within the brackets or parentheses. So, for this data:

$a[(b-c)+d] = [[(a\cdot b)-(a\cdot c)] + (a\cdot d)]$

We then simplify the expression by combining the variables we are multiplying together into expressions. For this data:

$[[(a\cdot b)-(a\cdot c)] + (a\cdot d)] = ab-ac+ad$

Be sure to keep all of your operations the same within the problem itself, unless the number being distributed is negative, which will then switch the signs with the brackets from positive to negative or negative to positive.

Therefore, our answer is ab-ac+ad .

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Example Question #14 : Polynomials

Solve the equation $x^{2} - 25x = 0$

Possible Answers:

Correct answer:

Explanation:

To answer this question, we are solving for the values of that make this equation true.

To this, we need to get on a side by itself so we can evaluate it. To do this, we first add 25x to both sides so that we can then begin to deal with the $x^{2}$ value. So, for this data:

$x^{2}-25x=0\rightarrow x^{2}=25x$

$x^{2}$ can also be written as $x\cdot x$ . Therefore:

$x^{2}=25x\rightarrow x\cdot x=25x$

Now we can divide both sides by and find the value of .

$\frac{x\cdot x}{x}=\frac{25x}{x}\rightarrow x=25$

Therefore, the answer to this question is x=25

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ACT Math : How to subtract polynomials

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Subtract Polynomials

Example Question #1 : How To Subtract Polynomials

Example Question #12 : Polynomials

Example Question #12 : Polynomials

Example Question #14 : Polynomials

All ACT Math Resources

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