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Math  /  Algebra

Question(x+3)(x3−4x+5)(x+3)(x3−4x+5)=□\begin{array}{r}(x+3)\left(x^{3}-4 x+5\right) \\ (x+3)\left(x^{3}-4 x+5\right)=\square\end{array}

Studdy Solution

STEP 1

1. We are given the expression (x+3)(x3−4x+5)(x+3)(x^3 - 4x + 5).
2. We need to expand this expression to find its equivalent polynomial form.

STEP 2

1. Apply the distributive property to expand the expression.
2. Simplify the resulting terms.
3. Combine like terms to get the final polynomial.

STEP 3

Apply the distributive property by multiplying each term in x+3x+3 with each term in x3−4x+5x^3 - 4x + 5:
(x+3)(x3−4x+5)=x(x3−4x+5)+3(x3−4x+5)(x+3)(x^3 - 4x + 5) = x(x^3 - 4x + 5) + 3(x^3 - 4x + 5)

STEP 4

Expand each part separately:
1. x(x3−4x+5)=x4−4x2+5xx(x^3 - 4x + 5) = x^4 - 4x^2 + 5x
2. 3(x3−4x+5)=3x3−12x+153(x^3 - 4x + 5) = 3x^3 - 12x + 15

STEP 5

Combine all the expanded terms:
x4−4x2+5x+3x3−12x+15x^4 - 4x^2 + 5x + 3x^3 - 12x + 15

STEP 6

Combine like terms:
- The x4x^4 term: x4x^4 - The x3x^3 term: 3x33x^3 - The x2x^2 term: −4x2-4x^2 - The xx terms: 5x−12x=−7x5x - 12x = -7x - The constant term: 1515
Putting it all together:
x4+3x3−4x2−7x+15x^4 + 3x^3 - 4x^2 - 7x + 15
The expanded form of the expression is:
x4+3x3−4x2−7x+15\boxed{x^4 + 3x^3 - 4x^2 - 7x + 15}

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