# The explanation of how to multiply polynomialswhen neither is monomial with an examplw: (2x^{2} + 4y^{3}) (3x^{3} + 4y^{2})

Question
Polynomial arithmetic
The explanation of how to multiply polynomialswhen neither is monomial with an examplw:
$$\displaystyle{\left({2}{x}^{{{2}}}\ +\ {4}{y}^{{{3}}}\right)}\ {\left({3}{x}^{{{3}}}\ +\ {4}{y}^{{{2}}}\right)}$$

2020-11-18
Step 1:
Distribute each term of the first polynomial to every term of the second polynomial.
$$\displaystyle{\left({2}{x}^{{{2}}}\ +\ {4}{y}^{{{3}}}\right)}\ {\left({3}{x}^{{{3}}}\ +\ {4}{y}^{{{2}}}\right)}={2}{x}^{{{2}}}\ {\left({3}{x}^{{{3}}}\right)}\ +\ {2}{x}^{{{2}}}\ {\left({4}{y}^{{{2}}}\right)}\ +\ {4}{y}^{{{3}}}\ {\left({3}{x}^{{{3}}}\right)}\ +\ {4}{y}^{{{3}}}$$
$$\displaystyle={6}{x}^{{{5}}}\ +\ {8}{x}^{{{2}}}\ {y}^{{{2}}}$$
$$\displaystyle+\ {12}{x}^{{{3}}}\ {y}^{{{3}}}\ +\ {16}{y}^{{{5}}}$$
Step 2: Combine like terms. In this problem, there are no line terms.
$$\displaystyle{6}{x}^{{{5}}}\ +\ {8}{x}^{{{2}}}\ {y}^{{{2}}}\ +\ {12}{x}^{{{3}}}\ {y}^{{{3}}}\ +\ {16}{y}^{{{5}}}$$
Conclusion:
Polynomial with Polynomial: To multiply a polynomial and a polynomial, use the distributive property until every term of one polynomial is mutiplied times every term of the other polynomial. Make sure that you simplify your answer by combining any like terms.
Example: $$\displaystyle{\left({2}{x}^{{{2}}}\ +\ {4}{y}^{{{3}}}\right)}\ {\left({3}{x}^{{{3}}}\ +\ {4}{y}^{{{2}}}\right)}={6}{x}^{{{5}}}\ +\ {8}{x}^{{{2}}}\ {y}^{{{2}}}\ +\ {12}{x}^{{{3}}}\ {y}^{{{3}}}\ +\ {16}{y}^{{{5}}}.$$

### Relevant Questions

Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae the same $$\displaystyle{P}{\left({x}\right)}={3}{x}^{{4}}-{5}{x}^{{3}}+{x}^{{2}}-{3}{x}+{5}{N}{S}{K}{Q}{\left({x}\right)}={\left({\left({\left({3}{x}-{5}\right)}{x}+{1}\right)}{x}-{3}\right)}{x}+{5}$$ Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial $$\displaystyle{R}{\left({x}\right)}={x}^{{5}}—{2}{x}^{{4}}+{3}{x}^{{3}}—{2}{x}^{{3}}+{3}{x}+{4}$$ in “nested” form, like the polynomial Q. Use the nested form to find R(3) in your head. Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value ofa polynomial using synthetic division?
DISCOVER: Nested Form of a Polynomial Expand Q to prove that the polynomials P and Q ae the same $$P(x) = 3x^{4} - 5x^{3} + x^{2} - 3x +5$$
$$Q(x) = (((3x - 5)x + 1)x 3)x + 5$$
Try to evaluate P(2) and Q(2) in your head, using the forms given. Which is easier? Now write the polynomial
R(x) =x^{5} - 2x^{4} + 3x^{3} - 2x^{2} + 3x + 4\) in “nested” form, like the polynomial Q. Use the nested form to find R(3) in your head.
Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value ofa polynomial using synthetic division?
Determine whether the following state-ments are true and give an explanation or counterexample.
a) All polynomials are rational functions, but not all rational functions are polynomials.
b) If f is a linear polynomial, then $$\displaystyle{f}\times{f}$$ is a quadratic polynomial.
c) If f and g are polynomials, then the degrees of $$\displaystyle{f}\times{g}$$ and $$\displaystyle{g}\times{f}$$ are equal.
d) To graph $$\displaystyle{g{{\left({x}\right)}}}={f{{\left({x}+{2}\right)}}}$$, shift the graph of f 2 units to the right.
(a)To calculate: The following equation {[(3x + 5)x + 4]x + 3} x + 1 = 3x^4 + 5x^3 + 4x^2 + 3x + 1 is an identity, (b) To calculate: The lopynomial P(x) = 6x^5 - 3x^4 + 9x^3 + 6x^2 -8x + 12 without powers of x as in patr (a).
Consider the following sequence.
$$s_{n} = 2n − 1$$
(a) Find the first three terms of the sequence whose nth term is given.
$$s_{1} =$$
$$s_{2} =$$
$$s_{3} =$$
(b) Classify the sequence as arithmetic, geometric, both, or neither. arithmetic, geometric bothneither
If arithmetic, give d, if geometric, give r, if both, give d followed by r. (If both, enter your answers as a comma-separated list. If neither, enter NONE.)
1. Explain with numerical examples what Real Numbers and Algebraic Expressions are. 2. Explain with numerical examples Factoring and finding LCMs (least common multiples). Explain factoring of a larger number. 3. Explain with numerical examples arithmetical operations (addition, subtraction, multiplication, division) with fractions 4, Explain with numerical examples arithmetical operations (addition, subtraction, multiplication, division) with percentages 5. Explain with numerical examples exponential notation 6. Explain with numerical examples order (precedence) of arithmetic operations 7. Explain with numerical examples the concept and how to find perimeter, area, volume, and circumference (use related formulas)
Given the following function: $$\displaystyle{f{{\left({x}\right)}}}={1.01}{e}^{{{4}{x}}}-{4.62}{e}^{{{3}{x}}}-{3.11}{e}^{{{2}{x}}}+{12.2}{e}^{{{x}}}-{1.99}$$ a)Use three-digit rounding frithmetic, the assumption that $$\displaystyle{e}^{{{1.53}}}={4.62}$$, and the fact that $$\displaystyle{e}^{{{n}{x}}}={\left({e}^{{{x}}}\right)}^{{{n}}}$$ to evaluate $$\displaystyle{f{{\left({1.53}\right)}}}$$ b)Redo the same calculation by first rewriting the equation using the polynomial factoring technique c)Calculate the percentage relative errors in both part a) and b) to the true result $$\displaystyle{f{{\left({1.53}\right)}}}=-{7.60787}$$
$$\displaystyle{f{{\left({x}\right)}}}={1.01}{e}^{{{4}{x}}}-{4.62}{e}^{{{3}{x}}}-{3.11}{e}^{{{2}{x}}}+{12.2}{e}^{{{x}}}$$
a) Use three-digit rounding frithmetic, the assumption that $$\displaystyle{e}^{{{1.53}}}={4.62}$$, and the fact that $$\displaystyle{e}^{{{n}{x}}}={\left({e}^{{{x}}}\right)}^{{{n}}}$$ to evaluate $$\displaystyle{f{{\left({1.53}\right)}}}$$
c) Calculate the percentage relative errors in both part a) and b) to the true result $$\displaystyle{f{{\left({1.53}\right)}}}=-{7.60787}$$
For Exercise, determine if the nth term of the sequence defines an arithmetic sequence, a geometric sequence, or neither. If the sequence is arithmetic, find the common difference d. If the sequence is geometric, find the common ratio r. $$a_{n} = 5 \pm \sqrt{2n}$$