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 and Polynomials

Adding and Subtracting Polynomials

6. Exponents and Polynomials

Adding and Subtracting Polynomials: Videos & Practice Problems

Video Lessons

Topic summary

Adding and subtracting polynomials involves combining like terms—terms with the same variable and exponent. When adding, parentheses can be removed without changing signs, but subtraction requires distributing the negative sign across all terms inside parentheses. Aligning terms vertically by degree, such as x2 for easier combination, helps organize operations. Understanding coefficients, exponents, and terms is essential for manipulating polynomials, including monomials and trinomials, and mastering these skills supports solving more complex algebraic expressions efficiently.

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.

Aligning like terms vertically is a helpful strategy, especially for more complex polynomials or word problems. By stacking terms with the same degree, such as x², x, and constants, you can add or subtract coefficients more easily. For example, adding 5x² + 2x + 3 and 2x² + 7x + 8 vertically aligns the terms and simplifies the process to get 7x² + 9x + 11.

In summary, the key to adding and subtracting polynomials lies in identifying and combining like terms, distributing negative signs when subtracting, and organizing terms effectively. Mastering these skills allows for efficient manipulation of polynomial expressions, which is fundamental in algebra and higher-level mathematics.

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 $- 4 x^{2} + 2 x + 8$ .

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

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

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

$x^2+14$

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 variable and exponent. First, remove any parentheses if the operation is addition, as the signs inside do not change. Then, identify terms with the same variable and exponent. For example, in adding $6x x^{2} + 3x$ and $2x - 8$ , you combine $3x$ and $2x$ to get $5x$ . Terms without like terms remain as they are. The final sum is $6x^2 + 5x - 8$ . This process ensures you only combine terms that are truly alike, maintaining the polynomial's structure.

The key step in subtracting polynomials is distributing the negative sign across all terms inside the parentheses of the polynomial being subtracted. For example, if you have $x^2 - 2x + 4 - (5x + 10)$ , you must distribute the minus sign to get $x^2 - 2x + 4 - 5x - 10$ . This changes the signs of each term inside the parentheses. After this, you combine like terms: $-2x - 5x = -7x$ and $4 - 10 = -6$ . The final expression is $x^2 - 7x - 6$ . This step is crucial to avoid errors in sign and ensure correct subtraction.

Aligning polynomials vertically by their like terms (same variable and exponent) helps organize the addition or subtraction process. For example, stacking $5x^2 + 2x + 3$ over $2x^2 + 7x + 8$ aligns the $x^2$ terms, the $x$ terms, and the constants in columns. This visual alignment makes it easier to combine like terms systematically: $5x^2 + 2x^2 = 7x^2$ , $2x + 7x = 9x$ , and $3 + 8 = 11$ . It reduces mistakes and speeds up 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 $-3x$ cannot be combined since one has an exponent of 2 and the other has an exponent of 1. Only terms with identical variables raised to the same power can be combined by adding or subtracting their coefficients. This distinction is fundamental when simplifying polynomials to ensure accuracy.

Constants are terms without variables and can be combined with other constants during addition or subtraction. For example, when adding $3$ and $-8$ , you simply perform the arithmetic operation: $3 - 8 = -5$ . In polynomials, constants are treated as like terms with each other but cannot be combined with terms containing variables. Always group constants together to simplify the polynomial expression effectively.