If p(x)=x^4-3x+2, then find the coefficient of the x^3 term in the polynomial (p(x))^3.
If you were to fully expand this, your unsimplified line would consist of 27 terms
- that is, any combination of terms in the first bracket times any term in the second bracket times any term in the third bracket
the only way to get x^3 is to have
-3 from the first times -3x from the second times -3x from the third, or
-27x^3
so ( p(x) )^3
= (x^4-3x+2)(x^4-3x+2)(x^4-3x+2)
now consider which combination can produce a term ending in x^3 ??
I don't understand what to do. Please clarify. Thank you.
To find the coefficient of the x^3 term in the polynomial (p(x))^3, we need to expand the expression (p(x))^3 and identify the coefficient of the x^3 term.
First, let's find p(x)^3. We can rewrite it as p(x) * p(x) * p(x).
Given p(x) = x^4 - 3x + 2.
To find p(x)^3, we'll substitute p(x) with its original expression and then expand using the distributive property:
p(x)^3 = (x^4 - 3x + 2) * (x^4 - 3x + 2) * (x^4 - 3x + 2).
Multiplying out this expression will give us the expanded form:
p(x)^3 = (x^4 - 3x + 2) * (x^4 - 3x + 2) * (x^4 - 3x + 2)
= (x^4 * x^4 * x^4) + (x^4 * x^4 * -3x) + (x^4 * x^4 * 2)
+ (-3x * x^4 * x^4) + (-3x * x^4 * -3x) + (-3x * x^4 * 2)
+ (2 * x^4 * x^4) + (2 * x^4 * -3x) + (2 * x^4 * 2)
+ (x^4 * -3x * x^4) + (x^4 * -3x * -3x) + (x^4 * -3x * 2)
+ (-3x * -3x * x^4) + (-3x * -3x * -3x) + (-3x * -3x * 2)
+ (2 * 2 * x^4) + (2 * 2 * -3x) + (2 * 2 * 2).
Now, simplify each term:
p(x)^3 = x^12 - 3x^9 + 2x^8 - 3x^9 + 9x^6 - 6x^5 + 2x^8 - 6x^5 + 4x^4
+ 2x^8 - 6x^5 + 4x^4 - 6x^5 + 18x^2 - 12x + 4x^4 - 12x + 8.
Finally, we can see that the x^3 term does not appear in the expanded form of p(x)^3. Therefore, the coefficient of the x^3 term is 0.