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Chapter 6: Problem 43

Add or subtract the polynomials. $$\left(a^{2}+8 a+5\right)-\left(a^{2}-3 a+2\right)$$

Short Answer

Expert verified
The result is \(11a + 3\).

Step by step solution

01

- Distribute the subtraction

Remove the parentheses and distribute the minus sign to each term in the second polynomial:\((a^{2} + 8a + 5) - (a^{2} - 3a + 2) = a^{2} + 8a + 5 - a^{2} + 3a - 2\)
02

- Combine like terms

Group and combine the like terms from the expression:\(a^{2} - a^{2} + 8a + 3a + 5 - 2\)
03

- Simplify

Simplify the expression by performing the arithmetic on the like terms:\(a^{2} - a^{2} = 0\)\(8a + 3a = 11a\)\(5 - 2 = 3\)\(0 + 11a + 3 = 11a + 3\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is crucial when performing operations like polynomial subtraction. It states that \ a(b + c) = ab + ac\. In the context of polynomials, you often use it to distribute a negative sign.

For example, in the expression \( (a^2 + 8a + 5) - (a^2 - 3a + 2)\), you distribute the minus sign across each term in the second polynomial: \[ (a^2 + 8a + 5) - (a^2 - 3a + 2) = a^2 + 8a + 5 - a^2 + 3a - 2. \] This step is critical because it changes the signs of the terms in the second polynomial, allowing you to combine like terms correctly.

Using the distributive property ensures that the operation maintains mathematical integrity and helps avoid common errors.

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Most popular questions from this chapter

Divide each polynomial by the binomial. $$\left(t^{2}-12 t+36\right) \div(t-6)$$

Divide the monomials. $$\frac{\left(4 u^{2} v^{5}\right)\left(15 u^{3} v\right)}{\left(12 u^{3} v\right)\left(u^{4} v\right)}$$

Divide the monomials. $$\frac{\left(-18 p^{4} q^{7}\right)\left(-6 p^{3} q^{8}\right)}{-36 p^{12} q^{10}}$$

Divide each polynomial by the binomial. $$\left(x^{2}-14 x+49\right) \div(x-7)$$

Divide each polynomial by the binomial. $$\left(64 x^{3}-27\right) \div(4 x-3)$$
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