Question

Find the product. Check your result by comparing a graph of the given expression with a graph of the product. $$ \left(3 x^{2}-8 x-1\right)(9 x+4) $$

Short answer

The product of \( (3 x^{2}-8 x-1) \) and \( (9 x+4) \) is \( 27 x^{3} -60 x^{2} -41x -4 \)

Step-by-step solution

Multiply the Polynomials

Using the distributive property, each term of the first polynomial is multiplied with each term of the second polynomial, i.e. multiply \((3x^2)\) with \( (9x) \) and \( (4) \), \((8x)\) with \( (9x) \) and \( (4) \), and \((-1)\) with \( (9x) \) and \( (4) \). This gives: \(27 x^{3} + 12 x^{2} - 72 x^{2} - 32 x - 9 x - 4\)

Combine Like Terms

Next, combine the terms that have the same variable raised to the same power. Simplify \(12 x^{2} - 72 x^{2}\) to get \(-60 x^{2}\) and \(-32 x -9 x\) to get \(-41 x\), resulting in the polynomial: \(27 x^{3} -60 x^{2} -41 x -4\)

Checking Result with Graphs

To verify the correctness of the result, graph both the expressions i.e. the given expression \((3 x^{2}-8 x-1)(9 x+4)\) and the obtained product \(27 x^{3} -60 x^{2} -41 x -4\). If both the graphs coincide with each other, then the multiplication is correctly performed.