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Chapter 10: Problem 70

Find the missing polynomial. \((9 y-3)+(\quad)=-4 y+7\).

Short Answer

Expert verified
The missing polynomial is P = -13y + 10.

Step by step solution

01

Understand the problem

We need to find a polynomial that, when added to (9y-3), results in the polynomial (-4y+7).
02

Set up the equation

Let's call the missing polynomial 'P'. Thus, the equation becomes (9y - 3) + P = -4y + 7.
03

Rearrange the equation

We can find P by isolating it. Subtract (9y - 3) from both sides: P = (-4y + 7) - (9y - 3).
04

Simplify

Distribute the subtraction across the polynomial: P = -4y + 7 - 9y + 3. Combine like terms: P = -13y + 10.

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Key Concepts

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

Understanding How to Solve Polynomials
Polynomials are mathematical expressions consisting of terms that can include variables, coefficients, and exponents. In this exercise, we're dealing with a problem that requires us to find a missing polynomial.
To solve a polynomial equation like \((9y - 3) + (\text{missing polynomial}) = -4y + 7\), you need to understand how to isolate variables and simplify expressions.
The steps are as follows:
  • Identify and comprehend the problem.
  • Set up the required equation.
  • Rearrange the equation to isolate the polynomial you're looking for.
  • Simplify the equation by performing the necessary operations.
By following these steps, you鈥檒l be able to systematically solve polynomial equations.
Combining Like Terms for Clarity
When working with polynomials, you will often encounter terms that can be combined. 'Like terms' are terms that have the same variable raised to the same power. For example, in the expression \(-4y + 7 - 9y + 3\), the like terms are \-4y\ and \-9y\.
Combining like terms involves summing their coefficients. In our example:
\begin{align*}-4y - 9y &= -13y,\ 7 + 3 &= 10.\right\begin{align*}By combining these terms, our simplified polynomial becomes \(-13y + 10\).
This process is crucial for simplifying polynomials and making the expressions easier to understand and solve.
Polynomial Subtraction: An Essential Skill
Subtracting polynomials requires distributing the subtraction across each term in the polynomial. For instance, in our example, when we subtract \(9y - 3\) from \(-4y + 7\), we need to apply the subtraction to both terms in the first polynomial.
This results in:
\begin{align*}-4y + 7 - 9y + 3\right\begin{align*}Distributing involves changing the sign of each term when subtracting, thus making the terms easier to combine. That鈥檚 how \7 - (-3)\ becomes \7 + 3\, and similarly for other terms.
Understanding how to subtract polynomials and distribute the subtraction will let you simplify complex polynomial expressions and isolate variables more effectively.

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