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Chapter 5: Problem 67

Find each product. In each case, neither factor is a monomial. $$(x+1)\left(x^{2}+2 x+3\right)$$

Short Answer

Expert verified
The product of \((x+1)\left(x^{2}+2 x+3\right)\) is \(x^3 + 3x^2 + 5x + 3.\)

Step by step solution

01

Distribute the first term in the first parenthesis

Multiply \(x\) from the first parenthesis with every term in the second parenthesis, i.e., \(x\) times \(x^2\), \(x\) times \(2x\) and \(x\) times \(3\). This gives: \(x\cdot x^2 + x\cdot 2x + x\cdot 3 = x^3 + 2x^2 + 3x.\)
02

Distribute the second term in the first parenthesis

Now, multiply 1 from the first parenthesis with every term in the second parenthesis, i.e., \(1\) times \(x^2\), \(1\) times \(2x\) and \(1\) times \(3\). This gives: \(1\cdot x^2 + 1\cdot 2x + 1\cdot 3 = x^2 + 2x + 3.\)
03

Combine like terms

Add together the results from Step 1 and Step 2 to get the final solution. When adding the polynomials, group terms with the same exponent of \(x\). This gives: \(x^3 + 2x^2 + 3x + x^2 + 2x + 3 = x^3 + 3x^2 + 5x + 3.\)

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