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

6 y 2 − 15 y − 2 6y^2-15y-2

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

7 x 3 + 7 x 2 + x y − 16 7x^3+7x^2+xy-16

7 x 3 + 7 x 2 + x y − 8 7x^3+7x^2+xy-8

7 x 3 + 7 x 2 + x y − 14 7x^3+7x^2+xy-14

7 x 3 + 7 x 2 + x y − 8 x 7x^3+7x^2+xy-8x

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

5 y 2 − 10 y + 13 5y^2-10y+13

− 7 y 2 + 10 y − 13 -7y^2+10y-13

5 y 2 − 10 y − 13 5y^2-10y-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 variables and exponents. 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 x², helps organize and simplify operations. Understanding coefficients, exponents, and terms is essential for manipulating polynomials, including monomials, binomials, and trinomials, ensuring accurate results in polynomial expressions and standard form.

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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 3x² and x cannot be combined 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. The 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 here, the -2x and -5x combine to -7x, and the constants 4 and -10 combine to -6, yielding the simplified polynomial x² - 7x - 6.

Another effective method for adding polynomials is to align like terms vertically, similar to arithmetic addition. For example, adding 5x² + 2x + 3 and 2x² + 7x + 8 by stacking them as:

\[\begin{array}{r}5x^{2} + 2x + 3 \\+ \quad 2x^{2} + 7x + 8 \\\hline\end{array}\]

allows you to add corresponding terms directly: 5x² + 2x² = 7x², 2x + 7x = 9x, and 3 + 8 = 11, resulting in 7x² + 9x + 11. This vertical alignment helps in clearly identifying and combining like terms.

Mastering the addition and subtraction of polynomials by combining like terms and correctly handling negative signs is fundamental for simplifying expressions and solving more complex algebraic problems.

Problem

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

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

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

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

$4 x^{2} + 15 x + 5$

Problem

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

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

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

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

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

Problem

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

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

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

$7 x^{3} + 7 x^{2} + x y - 14$

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

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 $6 y^{2} - 7 y + 9$ .

$5 y^{2} - 10 y + 13$

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

$7 y^{2} - 4 y + 5$

$5 y^{2} - 10 y - 13$

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 if the operation is addition, as the plus sign does not change the signs of the terms inside. Then, identify like terms by matching variables and their exponents. For example, $6x x^{2} + 3x$ and $2x - 8$ can be combined by adding the coefficients of like terms: $6 x^{2} + 2 x^{1} + 3 x^{1} - 8$ . Combine $3x$ and $2x$ to get $5x$ . The final sum is $6x x^{2} + 5x - 8$ . Remember, only like terms can be combined, so terms with different exponents or variables remain separate.

The key step in subtracting polynomials is distributing the negative sign across all terms inside the parentheses before combining like terms. For example, if you have $x x^{2} - 2x + 4 - (5x + 10)$ , you must 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 step is crucial because failing to distribute the negative sign will lead to incorrect results.

Aligning polynomials vertically by their like terms (same variables and exponents) helps organize the addition or subtraction process, making it easier to identify and combine like terms. For example, stacking $5x x^{2} + 2x + 3$ over $2x x^{2} + 7x + 8$ aligns the $x^2$ terms, the $x$ terms, and the constants in columns. Then, you add each column: $5x^2 + 2x^2 = 7x^2$ , $2x + 7x = 9x$ , and $3 + 8 = 11$ . This method 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, $5x x^{2}$ and $3x$ cannot be combined since one is $x^2$ and the other is $x^1$ . Only terms with identical variables raised to the same power can be combined by adding or subtracting their coefficients. This distinction is important to maintain the correct structure of the polynomial.

When polynomials are not written side by side, it is helpful to rewrite the expression by removing parentheses carefully and distributing any negative signs. For example, if you have $P = x x^{2} - 2x + 4 - (5x + 10)$ , distribute the minus sign to get $x x^{2} - 2x + 4 - 5x - 10$ . Then, combine like terms. If the polynomials are given in word problems or scattered form, rewriting them in a standard polynomial format or stacking them vertically can help organize the terms for easier addition or subtraction.