Add, subtract, and multiply polynomials; understand polynomial closure under these operations.
Multiply polynomials using organized partial products
Problem
Use organized partial products to multiply \((x^{2}~+~2x~+~1)(x~-~3)\). Give the partial product produced by \(x\), the partial product produced by \(-3\), and the simplified product.
Big Picture
What this problem is really about
Organized partial products let us handle the two terms in x minus three one at a time. Multiplying by x shifts every term up one degree, while multiplying by negative three changes every sign; once both rows are complete, we’ll align equal powers and add down the columns.
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