To solve the problem of subtracting the polynomial [tex]\(-7a^2 + 3a - 9\)[/tex] from [tex]\(5a^2 - 6a - 4\)[/tex], let's follow these steps:
1. Write down the polynomials:
- The first polynomial is [tex]\(5a^2 - 6a - 4\)[/tex].
- The second polynomial is [tex]\(-7a^2 + 3a - 9\)[/tex].
2. Set up the subtraction:
We need to subtract the second polynomial from the first. So we have:
[tex]\[
(5a^2 - 6a - 4) - (-7a^2 + 3a - 9)
\][/tex]
3. Distribute the negative sign:
Distribute the minus sign through the second polynomial:
[tex]\[
(5a^2 - 6a - 4) - (-7a^2 + 3a - 9) = 5a^2 - 6a - 4 + 7a^2 - 3a + 9
\][/tex]
Notice how every term in the second polynomial changes its sign.
4. Combine like terms:
- Combine the terms with [tex]\(a^2\)[/tex]: [tex]\( 5a^2 + 7a^2 = 12a^2 \)[/tex]
- Combine the terms with [tex]\(a\)[/tex]: [tex]\( -6a - 3a = -9a \)[/tex]
- Combine the constant terms: [tex]\(-4 + 9 = 5\)[/tex]
5. Write the result in standard form:
Put together all the combined terms:
[tex]\[
12a^2 - 9a + 5
\][/tex]
Thus, the final polynomial in standard form obtained by subtracting [tex]\(-7a^2 + 3a - 9\)[/tex] from [tex]\(5a^2 - 6a - 4\)[/tex] is:
[tex]\[
12a^2 - 9a + 5
\][/tex]
