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Chapter 6: Problem 45

Add or subtract the polynomials. $$\left(12 s^{2}-15 s\right)-(s-9)$$

Short Answer

Expert verified
The result of the polynomial operation is ewline = 12s^{2}-16s+9.

Step by step solution

01

Distribute the Negative Sign

Rewrite the subtraction as adding the negative. Distribute the negative sign to the terms inside the parentheses: ewlineewline=12s^2-15s-(s-9)ewline=12s^2-15s(-s+9)
02

Combine Like Terms

Combine the terms with the same variable powers: ewline12s^2-15s-s+9ewline=12s^2-16s+9

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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 a fundamental concept in algebra. It tells us how to multiply a single term by a polynomial. The property is written as \(a(b + c) = ab + ac\) and also applies to subtraction.
In this problem, we dealt with subtraction by distributing a negative sign across the terms inside the parenthesis.
Here鈥檚 a step-by-step look at how it was done:
  • Given problem: \((12s^2 - 15s) - (s - 9)\)
  • We rewrite the subtraction as adding the negative: \((12s^2 - 15s) + (-(s - 9))\)
  • Next, distribute the negative sign across the terms inside the parentheses: \( (12s^2 - 15s) + (-s + 9)\)
This gives us a new polynomial where each term has had its sign switched where necessary. The Distributive Property helps simplify polynomials before combining their like terms.
Combining Like Terms
Combining like terms is another key algebraic skill. It means to simplify an expression by merging terms that have the same variables raised to the same power. This step reduces complex expressions into simpler forms.
For our example:
  • After distributing the negative sign, we had: \(12s^2 - 15s - s + 9\)
  • Identify the like terms: Terms with \(s^2\), terms with \(s\), and constant terms.
  • Combine them:
    12\(s^2\) stays the same as there are no other \(s^2\) terms.
    \(-15s - s\) combines to give \(-16s\).
    \(9\) is a constant and has no like terms to combine with.
Putting this together, the simplified polynomial is \( 12s^2 - 16s + 9 \).
Remember, this process of combining like terms helps in reducing and solving polynomial equations much more manageable.
Algebraic Expressions
Algebraic expressions are mathematical phrases that can involve numbers, variables, and operation symbols. They represent values and can be manipulated using algebraic rules like the Distributive Property and combining like terms.
In this case, our original expression was \((12s^2 - 15s) - (s - 9)\).
Here is what we did:
  • Rewrote the expression to get rid of the parentheses: Transform subtraction into the addition of negatives.
  • Applied the distributive property: \((12s^2 - 15s) + (-s + 9)\)
  • Combined the like terms: \(12s^2 - 16s + 9\)
Each step allows us to simplify the given expression into its simplest form. This makes it easier to work with, especially when you need to solve equations later.
At a glance:
  • Variables are symbols that represent unknown values.
  • Constants are numbers on their own.
  • Coefficients are numbers in front of variables.
By carefully applying algebraic operations, you can transform and simplify algebraic expressions efficiently.

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Most popular questions from this chapter

Divide each polynomial by the binomial. $$\left(q^{2}+2 q+20\right) \div(q+6)$$

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

Simplify. (a) \((-7)^{-2}\) (b) \(-7^{-2}\) (c) \(\left(-\frac{1}{7}\right)^{-2}\) (d) \(-\left(\frac{1}{7}\right)^{-2}\)

Divide the monomials. $$\frac{65 a^{10} b^{8} c^{5}}{42 a^{7} b^{6} c^{8}}$$

Divide each polynomial by the monomial. $$\frac{20 y^{2}+12 y-1}{-4 y}$$
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