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Example 6.5

Multiply [tex]x^3 - x^2 + x - 2[/tex] by [tex]x^2 + 3x - 2[/tex].

The product is:
[tex] = x^3(x^2 + 3x - 2) - x^2(x^2 + 3x - 2) + x(x^2 + 3x - 2) - 2(x^2 + 3x - 2) [/tex]

Sagot :

GinnyAnsweridn
Sure, let's multiply [tex]\(x^3 - x^2 + x - 2\)[/tex] by [tex]\(x^2 + 3x - 2\)[/tex] step-by-step.

Given polynomials:
[tex]\[ P(x) = x^3 - x^2 + x - 2 \][/tex]
[tex]\[ Q(x) = x^2 + 3x - 2 \][/tex]

To find the product [tex]\(P(x) \cdot Q(x)\)[/tex], we need to multiply every term in the first polynomial by every term in the second polynomial and then combine like terms.

### Step-by-step multiplication:

1. Distribute [tex]\( x^3 \)[/tex] from [tex]\( P(x) \)[/tex]:
[tex]\[ x^3 \cdot (x^2 + 3x - 2) = x^5 + 3x^4 - 2x^3 \][/tex]

2. Distribute [tex]\( -x^2 \)[/tex] from [tex]\( P(x) \)[/tex]:
[tex]\[ -x^2 \cdot (x^2 + 3x - 2) = -x^4 - 3x^3 + 2x^2 \][/tex]

3. Distribute [tex]\( x \)[/tex] from [tex]\( P(x) \)[/tex]:
[tex]\[ x \cdot (x^2 + 3x - 2) = x^3 + 3x^2 - 2x \][/tex]

4. Distribute [tex]\( -2 \)[/tex] from [tex]\( P(x) \)[/tex]:
[tex]\[ -2 \cdot (x^2 + 3x - 2) = -2x^2 - 6x + 4 \][/tex]

### Combine the results:

[tex]\[ x^5 + 3x^4 - 2x^3 \][/tex]
[tex]\[ -x^4 - 3x^3 + 2x^2 \][/tex]
[tex]\[ x^3 + 3x^2 - 2x \][/tex]
[tex]\[ -2x^2 - 6x + 4 \][/tex]

### Combine like terms:

1. Combine [tex]\( x^5 \)[/tex] term:
[tex]\[ x^5 \][/tex]

2. Combine [tex]\( x^4 \)[/tex] terms:
[tex]\[ 3x^4 - x^4 = 2x^4 \][/tex]

3. Combine [tex]\( x^3 \)[/tex] terms:
[tex]\[ -2x^3 - 3x^3 + x^3 = -4x^3 + x^3 = -4x^3 \][/tex]

4. Combine [tex]\( x^2 \)[/tex] terms:
[tex]\[ 2x^2 + 3x^2 - 2x^2 - 2x^2 = 3x^2 - 2x^2 = -4x^2 + 3x^2 = x^2 \][/tex]

5. Combine [tex]\( x \)[/tex] terms:
[tex]\[ -2x - 6x = -8x \][/tex]

6. Combine constant terms:
[tex]\[ 4 \][/tex]

So, the resulting polynomial after multiplying [tex]\(x^3 - x^2 + x - 2\)[/tex] by [tex]\(x^2 + 3x - 2\)[/tex] is:

[tex]\[ x^5 + 2x^4 - 4x^3 + x^2 - 8x + 4 \][/tex]

Therefore, the product is:

[tex]\[ (x^3 - x^2 + x - 2)(x^2 + 3x - 2) = x^5 + 2x^4 - 4x^3 + x^2 - 8x + 4 \][/tex]
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