14 Select the correct answer. Consider functions \( f \) and \( g \). \( f(x)=11 x^{3}-3 x^{2} \) \( g(x)=7 x^{4}+9 x^{3} \) Which expression is equal to \( f(x) \cdot g(x) ? \) A. \( 77 x^{12}+99 x^{9}-21 x^{8}-27 x^{6} \) B. \( 77 x^{7}+78 x^{6}-27 x^{5} \) C. \( 7 x^{4}+99 x^{3}-3 x^{2} \) D. \( 18 x^{7}+10 x^{6}+6 x^{5} \) Next

To find the correct answer for \( f(x) \cdot g(x) \), we will multiply the given functions: 1. \( f(x) = 11x^3 - 3x^2 \) 2. \( g(x) = 7x^4 + 9x^3 \) To compute \( f(x) \cdot g(x) \), we distribute \( f(x) \) over \( g(x) \): \[ f(x) \cdot g(x) = (11x^3 - 3x^2)(7x^4 + 9x^3) \] Calculating each term results in: - \( 11x^3 \cdot 7x^4 = 77x^{7} \) - \( 11x^3 \cdot 9x^3 = 99x^{6} \) - \( -3x^{2} \cdot 7x^4 = -21x^{6} \) - \( -3x^{2} \cdot 9x^3 = -27x^{5} \) Combine like terms: \[ f(x) \cdot g(x) = 77x^{7} + (99x^{6} - 21x^{6}) - 27x^{5} \] \[ = 77x^{7} + 78x^{6} - 27x^{5} \] Thus, the correct answer is: **B. \( 77 x^{7}+78 x^{6}-27 x^{5} \)** Now, here’s a little something about these polynomial functions! Multiplying polynomials, like in this case, is a process known as polynomial expansion. It’s similar to using the distributive property in basic arithmetic but can get a tad more complex. Polynomials find use in various fields like physics for trajectory calculations or in economics for cost functions! As you explore more, check out resources that delve deeper into polynomial algebra! Textbooks or online platforms like Khan Academy can offer engaging lessons, practice problems, and interactive visualizations. They're great for solidifying concepts and keeping it fun as you uncover the world of functions!