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Answer :
To solve the given expression:
[tex]\[
(2x^2 + 7x^3 - 5x^4) - (4x^4 + 2x^2 - 5x^3)
\][/tex]
we need to subtract the second polynomial from the first. Here's how you can do it step-by-step:
1. Identify like terms: Look for terms with the same degree of [tex]\(x\)[/tex] in both polynomials.
2. Subtract coefficients of like terms:
- For [tex]\(x^4\)[/tex]: The coefficient from the first polynomial is [tex]\(-5\)[/tex], and from the second polynomial is [tex]\(4\)[/tex]. Subtracting these gives: [tex]\(-5 - 4 = -9\)[/tex].
- For [tex]\(x^3\)[/tex]: The coefficient from the first polynomial is [tex]\(7\)[/tex], and from the second is [tex]\(-5\)[/tex]. Subtracting these gives: [tex]\(7 - (-5) = 7 + 5 = 12\)[/tex].
- For [tex]\(x^2\)[/tex]: The coefficient from the first polynomial is [tex]\(2\)[/tex], and from the second is [tex]\(2\)[/tex] as well. Subtracting these gives: [tex]\(2 - 2 = 0\)[/tex].
3. Write the resulting polynomial using the new coefficients:
[tex]\[
-9x^4 + 12x^3 + 0x^2
\][/tex]
Since the [tex]\(x^2\)[/tex] term has a coefficient of zero, you can omit it. This makes the final answer:
[tex]\[
-9x^4 + 12x^3
\][/tex]
This matches one of the choices given:
[tex]\[
\boxed{-9x^4 + 12x^3}
\][/tex]
[tex]\[
(2x^2 + 7x^3 - 5x^4) - (4x^4 + 2x^2 - 5x^3)
\][/tex]
we need to subtract the second polynomial from the first. Here's how you can do it step-by-step:
1. Identify like terms: Look for terms with the same degree of [tex]\(x\)[/tex] in both polynomials.
2. Subtract coefficients of like terms:
- For [tex]\(x^4\)[/tex]: The coefficient from the first polynomial is [tex]\(-5\)[/tex], and from the second polynomial is [tex]\(4\)[/tex]. Subtracting these gives: [tex]\(-5 - 4 = -9\)[/tex].
- For [tex]\(x^3\)[/tex]: The coefficient from the first polynomial is [tex]\(7\)[/tex], and from the second is [tex]\(-5\)[/tex]. Subtracting these gives: [tex]\(7 - (-5) = 7 + 5 = 12\)[/tex].
- For [tex]\(x^2\)[/tex]: The coefficient from the first polynomial is [tex]\(2\)[/tex], and from the second is [tex]\(2\)[/tex] as well. Subtracting these gives: [tex]\(2 - 2 = 0\)[/tex].
3. Write the resulting polynomial using the new coefficients:
[tex]\[
-9x^4 + 12x^3 + 0x^2
\][/tex]
Since the [tex]\(x^2\)[/tex] term has a coefficient of zero, you can omit it. This makes the final answer:
[tex]\[
-9x^4 + 12x^3
\][/tex]
This matches one of the choices given:
[tex]\[
\boxed{-9x^4 + 12x^3}
\][/tex]
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