Chapter 12

Without solving the equation, decide how many solutions it has. $$ \left(x^{4}+2\right)\left(3+x^{2}\right)=0 $$

Write the polynomials in exercises in standard form $$a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}$$ What are the values of the coefficients \(a_{0}, a_{1}, \ldots, a_{n} ?\) Give the degree of the polynomial. $$ 1+x+\frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{24}+\frac{x^{5}}{120} $$

$$ \begin{aligned} &\text { Find the solutions of }\\\ &\left(x^{2}-a^{2}\right)(x+1)=0, \quad a \text { a constant } \end{aligned} $$

Without expanding, what is the constant term of $$ (x+2)(x+3)(x+4)(x+5)(x+6) ? $$

For what value(s) of the constant \(a\) does \(\left(x^{2}-a^{2}\right)(x+1)=0\) have exactly two solutions?

For what values of \(a\) does the equation have a solution in \(x\) ? $$ x^{2}-a=0 $$

Without expanding, what is the leading term of $$ (2 s+5)(3 s+1)(s-10) ? $$

What is the degree and leading coefficient of the polynomial \(r(x)=4 ?\)

For what values of \(a\) does the equation have a solution in \(x\) ? $$ 2 x^{2}+a=0 $$

Refer to Example 2 on page 379 about the value of annual gifts to Elliot growing at an annual growth factor of \(x=1+r,\) where \(r\) is the annual interest rate. Suppose that the first three gifts were \(\$ 1000, \$ 500,\) and \(\$ 750 .\) (a) If \(r=5 \%\), what is the total value of the investments on his \(17^{\text {th }}\) birthday? On his \(18^{\text {th }}\) birthday? (b) Write polynomial expression in \(x\) for the value on his \(17^{\text {th }}\) and \(18^{\text {th }}\) birthday.

For what values of \(a\) does the equation have a solution in \(x\) ? $$ a x^{2}-5=0 $$

For what values of \(a\) does the equation have a solution in \(x\) ? $$ \left(a x^{2}+1\right)(x-a)=0 $$

Refer to Example 2 on page 379 about the value of annual gifts to Elliot growing at an annual growth factor of \(x=1+r,\) where \(r\) is the annual interest rate. The total value of his investments on his \(20^{\text {th }}\) birthday is $$1000 x^{5}+500 x^{4}+750 x^{3}+1200 x+650$$ (a) What were the gifts on his \(18^{\text {th }}, 19^{\text {th }}\) and \(20^{\text {th }}\) birthdays? (b) Evaluate the polynomial in part (a) for \(x=\) \(1.05,1.06,1.07 .\) What do these values tell you about the investment?

For what values of \(a\) does the equation have a solution in \(x\) ? $$ a^{2}+a x^{2}=0 $$

For what values of \(a\) does the equation have a solution in \(x\) ? $$ \left(x^{2}+a\right)\left(x^{2}-a\right)=0 $$

What is the degree of the resulting polynomial? The product of a quadratic and a linear polynomial.

For what values of \(a\) does the equation have a solution in \(x\) ? $$ x^{3}+a=0 $$

What is the degree of the resulting polynomial? The product of two linear polynomials.

For what values of \(a\) does the equation have a solution in \(x\) ? $$ x^{4}+5 a=0 $$

What is the degree of the resulting polynomial? The sum of a degree 8 polynomial and a degree 4 polynomial.

For what values of \(a\) does the equation have a solution in \(x\) ? $$ a-x^{5}=0 $$

What is the value of $$5(x-1)(x-2)+2(x-1)(x-3)-4(x-2)(x-3)$$ when \(x=3 ?\)

Find approximate solutions to $$ 3 x^{3}-2 x^{2}-6 x+4=0 $$ by graphing the polynomial.

What values of the constants \(A, B,\) and \(C,\) will make \(A(x-1)(x-2)+B(x-1)(x-3)-C(x-2)(x-3)\) have the value 7 when \(x=3 ?\)

Consider the polynomial \(x^{5}-3 x^{4}+4 x^{3}-2 x+1\) (a) What is the value of the polynomial when \(x=4\) ? (b) If \(a\) is the answer you found in part (a), show that \(x-4\) is a factor of \(x^{5}-3 x^{4}+4 x^{3}-2 x+1-a\)

Evaluate the expressions in Problems \(51-54\) given that \(f(x)=2 x^{3}+3 x-3, \quad g(x)=3 x^{2}-2 x-4\) \(h(x)=f(x) g(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}\) $$ n $$

Use the identity \((x-1)\left(x^{4}+x^{3}+x^{2}+x+1\right)=x^{5}-1\) to show that \(8^{5}-1\) is divisible by 7 .

Evaluate the expressions in Problems \(51-54\) given that \(f(x)=2 x^{3}+3 x-3, \quad g(x)=3 x^{2}-2 x-4\) \(h(x)=f(x) g(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}\) $$ a_{n-1} $$

Consider the polynomial \(p(x)=(x-k)^{n},\) where \(k\) is a constant and \(n\) is a positive integer. (a) If \(n\) is even explain why the graph of \(p(x)\) is never below the \(x\) -axis. (b) If \(n\) is odd explain why the graph of \(p(x)\) is below the \(x\) -axis for \(xk\).

Evaluate the expressions in Problems \(51-54\) given that \(f(x)=2 x^{3}+3 x-3, \quad g(x)=3 x^{2}-2 x-4\) \(h(x)=f(x) g(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}\) $$ a_{0} $$

Evaluate the expressions in Problems \(51-54\) given that \(f(x)=2 x^{3}+3 x-3, \quad g(x)=3 x^{2}-2 x-4\) \(h(x)=f(x) g(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{0}\) $$ h(1) $$

If the following product of two polynomials, $$\left(3 t^{2}-7 t-2\right)\left(4 t^{3}-3 t^{2}+5\right)$$ is written in standard form, what are the constant and leading terms?

State the given quantities if \(p(x)\) is a polynomial of degree 5 with constant term 3 , and \(q(x)\) is a polynomial of degree 8 with constant term -2. The constant term of \(p(x) q(x)\).

State the given quantities if \(p(x)\) is a polynomial of degree 5 with constant term 3 , and \(q(x)\) is a polynomial of degree 8 with constant term -2. The degree of \(p(x)+q(x)\).

State the given quantities if \(p(x)\) is a polynomial of degree 5 with constant term 3 , and \(q(x)\) is a polynomial of degree 8 with constant term -2. The degree of \(p(x) q(x)\).

State the given quantities if \(p(x)\) is a polynomial of degree 5 with constant term 3 , and \(q(x)\) is a polynomial of degree 8 with constant term -2. The constant term of \(p(x)-2 q(x)\).

State the given quantities if \(p(x)\) is a polynomial of degree 5 with constant term 3 , and \(q(x)\) is a polynomial of degree 8 with constant term -2. The degree of \(p(x)^{3} q(x)^{2}\).

Given that \(\left(3 x^{3}+a\right)\left(2 x^{b}+3\right)=6 x^{7}+c x^{4}+d x^{3}+3\), find possible values for the constants \(a, b, c,\) and \(d\).

Find the constants \(r, s, p,\) and \(q\) if multiplying out the polynomial \(\left(r x^{5}+2 x^{4}+3\right)\left(2 x^{3}-s x^{2}+p\right)\) gives \(6 x^{8}-11 x^{7}-10 x^{6}-12 x^{5}-8 x^{4}+q x^{3}-15 x^{2}-12 .\)

Give polynomials satisfying the given conditions if possible, or say why it is impossible to do so. Two polynomials whose product has degree \(9 .\)

Give the value of \(a\) that makes the statement true. The degree of \((t-1)^{3}+a(t+1)^{3}\) is less than 3

Give the value of \(a\) that makes the statement true. The coefficient of \(t\) in \(t\left(a+(t+1)^{10}\right)\) is zero.

Give the value of \(a\) that makes the statement true. The constant term of \((t+2)^{2}(t-a)^{2}\) is 9 .

Suppose that two polynomials \(p(x)\) and \(q(x)\) have constant term \(1,\) the coefficient of \(x\) in \(p(x)\) is \(a\) and the coefficient of \(x\) in \(q(x)\) is \(b\). What is the coefficient of \(x\) in \(p(x) q(x) ?\)

Find the product of \(5 x^{2}-3 x+1\) and \(10 x^{3}-3 x^{2}-1\).

Find the leading term and constant term of (a) \((x-1)^{2}\) (b) \((x-1)^{3}\) (c) \((x-1)^{4}\)

What is the coefficient of \(x^{n-1}\) in \((x+1)^{n}\) for \(n=2,3\) and \(4 ?\)