Question
simplify the polynominal expression 7x^2 + 2x -9 /7x + 9 (x + 2 ) (x - 3)
Answers
GPT-4o mini
To simplify the polynomial expression \(\frac{7x^2 + 2x - 9}{7x + 9} (x + 2)(x - 3)\), we'll break it down step by step.
1. **Factor the numerator \(7x^2 + 2x - 9\)** if possible.
To factor this quadratic expression, we look for two numbers that multiply to \(7 \cdot (-9) = -63\) and add up to \(2\).
The numbers \(9\) and \(-7\) work (since \(9 + (-7) = 2\)).
We can then split the middle term:
\[
7x^2 + 9x - 7x - 9 = (7x^2 - 7x) + (9x - 9)
\]
Factor by grouping:
\[
= 7x(x - 1) + 9(x - 1) = (7x + 9)(x - 1)
\]
Hence, \(7x^2 + 2x - 9 = (7x + 9)(x - 1)\).
2. **Rewrite the expression** with the factored form:
\[
\frac{(7x + 9)(x - 1)}{7x + 9} \cdot (x + 2)(x - 3)
\]
3. **Cancel out \(7x + 9\)** in the numerator and denominator (as long as \(7x + 9 \neq 0\)):
\[
(x - 1)(x + 2)(x - 3)
\]
4. **Final simplified expression**:
The simplified expression is:
\[
(x - 1)(x + 2)(x - 3)
\]
This yields the final result of the simplification:
\[
(x - 1)(x + 2)(x - 3)
\]
This is the simplified polynomial expression.
1. **Factor the numerator \(7x^2 + 2x - 9\)** if possible.
To factor this quadratic expression, we look for two numbers that multiply to \(7 \cdot (-9) = -63\) and add up to \(2\).
The numbers \(9\) and \(-7\) work (since \(9 + (-7) = 2\)).
We can then split the middle term:
\[
7x^2 + 9x - 7x - 9 = (7x^2 - 7x) + (9x - 9)
\]
Factor by grouping:
\[
= 7x(x - 1) + 9(x - 1) = (7x + 9)(x - 1)
\]
Hence, \(7x^2 + 2x - 9 = (7x + 9)(x - 1)\).
2. **Rewrite the expression** with the factored form:
\[
\frac{(7x + 9)(x - 1)}{7x + 9} \cdot (x + 2)(x - 3)
\]
3. **Cancel out \(7x + 9\)** in the numerator and denominator (as long as \(7x + 9 \neq 0\)):
\[
(x - 1)(x + 2)(x - 3)
\]
4. **Final simplified expression**:
The simplified expression is:
\[
(x - 1)(x + 2)(x - 3)
\]
This yields the final result of the simplification:
\[
(x - 1)(x + 2)(x - 3)
\]
This is the simplified polynomial expression.