Add or subtract the polynomials.(3x2+9x+15)−(−$$x2+6x+10)(3x^2$+9x+15)-(-x^2+6x+10) $$

$$4x2+15x+54x^2+15x+54x2+15x+5$$

Add or subtract the polynomials.(2y2−9y+5)+(6y+4y2−$$7)(2y^2$-9y+5)+(6y+4y^2-7) $$

6y2−3y−$$26y^2-3y-26y2$$−3y−2

Add or subtract the polynomials.(7x3+xy−8)+(7x2−$$8)(7x^3$+xy-8)+(7x^2-8) $$

7x3+7x2+xy−$$167x^3$+7x^2+xy-167x3+7x2+xy$$−16

7x3+7x2+xy−$$87x^3$+7x^2+xy-8$$

7x3+7x2+xy−$$147x^3$+7x^2+xy-147x3+7x2+xy$$−14

7x3+7x2+xy−$$8x7x^3$+7x^2+xy-8x7x3+7x2+xy$$−8x

Subtract y2+3y−$$4y^2+3y-4 $$from 6y2−$$7y+96y^2-7y+9. $$

5y2−$$10y+135y^2-10y+135y2$$−10y+13

−7y2+10y−$$13-7y^2+10y-13$$−7y2+10y−13

7y2−$$4y+57y^2-4y+57y2$$−4y+5

5y2−10y−$$135y^2-10y-135y2$$−10y−13

6. Exponents, Polynomials, and Polynomial Functions

Adding and Subtracting Polynomials

6. Exponents, Polynomials, and Polynomial Functions

Adding and Subtracting Polynomials: Videos & Practice Problems

Video Lessons

Topic summary

Adding and subtracting polynomials involves combining like terms—terms with the same variables and exponents. When subtracting, distribute the negative sign across the polynomial inside parentheses before combining. Aligning terms vertically by degree, such as x2 for x squared, helps organize operations. This process applies to monomials, binomials, and trinomials, emphasizing the importance of coefficients and exponents. Mastery of these operations is essential for manipulating polynomials in standard form and understanding concepts like degree, numerical coefficient, and terms in multivariable polynomials.

Concept

Adding and Subtracting Polynomials

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Adding and Subtracting Polynomials Video Summary

Adding and subtracting polynomials involves combining like terms, which are terms that have the same variables raised to the same exponents. For example, terms like 5x² and -3x² can be combined because they share the variable x raised to the power of 2, but 3x cannot be combined with 5x² since the exponents differ.

When adding polynomials, you can remove parentheses without changing the signs of the terms. For instance, adding (6x² + 3x) and (2x - 8) simplifies to 6x² + 3x + 2x - 8. Then, by combining like terms, the 3x and 2x add up to 5x, resulting in the simplified polynomial 6x² + 5x - 8.

Subtracting polynomials requires careful attention to the negative sign in front of parentheses. This negative sign must be distributed to each term inside the parentheses, effectively flipping their signs. For example, subtracting (5x + 10) from x² - 2x + 4 becomes x² - 2x + 4 - 5x - 10 after distribution. Combining like terms then yields x² - 7x - 6.

Another useful method for adding polynomials is to align like terms vertically, similar to how numbers are added in arithmetic. For example, when adding 5x² + 2x + 3 and 1x² + 7x + 8, stacking the terms by degree makes it easier to combine: the x² terms add to 6x², the x terms add to 9x, and the constants add to 11, resulting in 6x² + 9x + 11.

Mastering the addition and subtraction of polynomials by combining like terms and properly handling subtraction through distribution is essential for simplifying expressions and solving algebraic problems efficiently.

Problem

Add or subtract the polynomials.
$(3 x^{2} + 9 x + 15) - (- x^{2} + 6 x + 10)$

$4 x^{2} + 3 x + 25$

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

$4x^2+3x+5$

$4x^2+15x+5$

Problem

Add or subtract the polynomials.
$(2 y^{2} - 9 y + 5) + (6 y + 4 y^{2} - 7)$

$6 y^{2} - 3 y + 2$

$6 y^{2} - 15 y - 2$

$6 y^{2} - 3 y + 12$

$6y^2-3y-2$

Problem

Add or subtract the polynomials.
$(7 x^{3} + x y - 8) + (7 x^{2} - 8)$

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

$7x^3+7x^2+xy-16$

$7x^3+7x^2+xy-14$

$7x^3+7x^2+xy-8x$

Problem

Find the sum of $5 x^{2} - 2 x + 6$ and $- 4 x^{2} + 2 x + 8$ .

$x^{2} - 6 x + 16$

$x^{2} - 6 x + 14$

$x^{2} + 4 x + 14$

$x squared plus 14$

Problem

Subtract $y^{2} + 3 y - 4$ from $6 y^{2} - 7 y + 9$ .

$5y^2-10y+13$

$-7y^2+10y-13$

$7y^2-4y+5$

$5y^2-10y-13$

Here's what students ask on this topic:

Adding polynomials involves combining like terms, which are terms with the same variables and exponents. First, remove any parentheses by dropping them if there is a plus sign before the polynomial. Then, identify terms that have the same variable and exponent. For example, in $6x x^{2} + 3x$ and $+ 2x - 8$ , you combine $3x$ and $2x$ because they are like terms, resulting in $5x$ . Terms without like counterparts remain as they are. The final sum is the polynomial with all like terms combined, such as $6x x^{2} + 5x - 8$ . This method works for monomials, binomials, and trinomials alike.

When subtracting polynomials, the key step is to distribute the negative sign across the polynomial inside the parentheses before combining like terms. For example, if you have $x x^{2} - 2x + 4 - (5x + 10)$ , distribute the minus sign to get $x x^{2} - 2x + 4 - 5x - 10$ . Then, combine like terms: $x x^{2}$ stays the same, $-2x - 5x = -7x$ , and $4 - 10 = -6$ . The final expression is $x x^{2} - 7x - 6$ . This ensures accurate subtraction by correctly handling the signs.

Aligning polynomials vertically by their like terms—such as $x^{2}$ , $x$ , and constants—helps organize the addition or subtraction process. This method places terms with the same variables and exponents directly above or below each other, making it easier to identify and combine like terms. For example, stacking $5x^2 + 2x + 3$ over $2x^2 + 7x + 8$ allows you to add corresponding terms directly: $5x^2 + 2x^2 = 7x^2$ , $2x + 7x = 9x$ , and $3 + 8 = 11$ . This vertical alignment reduces errors and simplifies calculations, especially with longer polynomials.

No, you cannot combine terms that have the same variable but different exponents because they are not like terms. For example, $5x^2$ and $3x$ both contain the variable $x$ , but since their exponents differ ( $2$ and $1$ respectively), they must remain separate. Only terms with identical variables raised to the same power can be combined by adding or subtracting their coefficients. This distinction is crucial when simplifying polynomials to ensure accuracy.

When subtracting a polynomial inside parentheses, distribute the negative sign to each term inside before combining like terms. For example, in $x^2 - 2x + 4 - (5x + 10)$ , distribute the minus sign to get $x^2 - 2x + 4 - 5x - 10$ . This changes the signs of all terms inside the parentheses. After distribution, combine like terms: $-2x - 5x = -7x$ and $4 - 10 = -6$ . The final expression is $x^2 - 7x - 6$ . This step is essential to avoid sign errors in subtraction.