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Chapter 5: Problem 45

In the following exercises, subtract the polynomials. \(\left(12 s^{2}-15 s\right)-(s-9)\)

Short Answer

Expert verified
The result is \(12s^2 - 16s + 9\).

Step by step solution

01

Distribute the subtraction

Subtraction of polynomials involves distributing the negative sign through the expression being subtracted. Rewrite the expression: \( (12s^2 - 15s) - (s - 9) = 12s^2 - 15s - s + 9 \)
02

Combine like terms

Combine the like terms from the expression obtained in Step 1: \(12s^2 - 15s - s + 9 = 12s^2 - 16s + 9 \)

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Key Concepts

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

Distributing Negative Signs
When subtracting polynomials, you first need to distribute the negative sign through the polynomial you are subtracting. This process changes the sign of every term in that polynomial. For example, if you have to subtract \( (s - 9) \) from \( (12s^2 - 15s) \), you rewrite it by changing the signs of each term inside the parentheses: \((12s^2 - 15s) - (s - 9) \) becomes \( 12s^2 - 15s - s + 9 \). Let's break it down:
  • The first polynomial \( 12s^2 - 15s \) remains the same.

  • The second polynomial \( (s - 9) \) will change its signs to \( -s + 9 \).

  • This happens because you're technically multiplying every term in the polynomial by \( -1 \).
This step is crucial for correctly simplifying the expression in polynomial subtraction.
Combining Like Terms
After distributing negative signs, the next step in polynomial subtraction is to combine like terms. Like terms are terms that have the same variables raised to the same power. Consider the expression \( 12s^2 - 15s - s + 9 \):
  • Identify and group all like terms:

  • \( -15s \) and \( -s \) are like terms because they both contain the variable \( s \) raised to the first power.

  • Combine these terms: \( -15s - s \) simplifies to \( -16s \).
Now, your expression looks like this: \( 12s^2 - 16s + 9 \).
Remember: Always add or subtract coefficients of like terms while keeping the variable part unchanged. This gives you a simpler polynomial and ensures you haven't lost or changed any information during subtraction.
Polynomial Subtraction
Polynomial subtraction follows a systematic process to ensure accuracy:
  • First, distribute the negative sign through the polynomial being subtracted.

  • Then, combine like terms to simplify the expression.
Using our example, we had to subtract \( (s - 9) \) from \( (12s^2 - 15s) \). After distributing the negative sign, it converted to \(12s^2 - 15s - s + 9 \). By combining like terms, we simplified it to \( 12s^2 - 16s + 9 \).
This process ensures that we accurately account for all terms in both polynomials and simplifies the expression correctly. Keep practicing these steps with different problems to master polynomial subtraction!

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Most popular questions from this chapter

Multiply the binomials using (a) the Distributive Property; (b) the FOIL method; (c) the Vertical Method. $$ (y+9)(y+3) $$

Multiply each pair of conjugates using the Product of Conjugates Pattern. $$ \left(15 m^{2}-8 n^{4}\right)\left(15 m^{2}+8 n^{4}\right) $$

Find each product. $$ (3 x-11 y)(3 x-11 y) $$

Multiply. (a) \(-8 x\left(x^{2}+2 x-15\right)\) (b) \(5 p q^{3}\left(p^{2}-2 p q+6 q^{2}\right)\)

Find each product. $$ (2 x+y)(x-2 y) $$
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