1.11A Equivalent representations of polynomial and rational expressions
For each polynomial:
Identify the leading coefficient.
Determine the degree of the polynomial.
x^4-5x^2+4
Rewrite the polynomial expressions in factored form:
f\left(x\right)=x^4+6x^3+11x^2+6x+1
f\left(x\right)=x^3-5x^2+6x
f\left(x\right)=x^3-2x^2-x+2
f\left(x\right)=x^3+x^2-x-1
Determine the zeros of the polynomial.
f\left(x\right)=(x+2)(x+3)(x+4)
Rewrite the polynomial expressions in standard form:
f\left(x\right)=(x^2+3x+2)(x^2-2x+1)
f\left(x\right)=(x^2+2x+3)(x^2+5x+6)
Christa wants to test whether various linear expressions divide exactly into P \left( x \right), or whether they leave a remainder. For each linear expression below, what is the value of x that needs to be substituted into P \left( x \right) to find the remainder?
x + 3
8 - x
5 + 4 x
6 - x
When performing long division with polynomials, we often express the function f(x) in the format of f(x) = g(x)q(x) + r(x). In this equation, what do the variables g, q, and r represent?
A rectangle has a width of 4x+5 and an area of 5 x^{3} + 7 x^{2} - 18 x - 8.
Find a polynomial expression for its length.
For the space station, an engineer has designed a new rectangular solar panel that has an area of \left( 24 x^{3} - 24 x^{2} + 10 x - 2\right) \text{ ft}^2. The length of the solar panel is \left( 6 x^{2} - 3 x + 1\right)\text{ ft}.
Find the width of the solar panel.
Consider the polynomial function f(x) = x^3 - 7x^2 + 14x - 8. Use information from different analytic representations of the function to answer the following questions:
Determine the degree of the polynomial.
Find the x-intercepts. How do these relate to the factors of the polynomial?
Determine the leading coefficient of the polynomial. How is this value reflected in the graph of the function?
Does the given polynomial form reveal any symmetries of the function? If so, what are they? If not, explain why.
How does the degree of the polynomial influence the general shape of the graph?
Find the value of k for each of the following:
The remainder when 3 x^{3} - 2 x^{2} - 4 x + k is divided by x - 3 is 47.
The remainder when 3 x^{3} + 4 x^{2} + k x - 3 is divided by x + 1 is - 5.
The remainder when 3 x^{3} + 4 x^{2} + 4 x + k is divided by x - 2 is 52.
The remainder when 4 x^{3} - 2 x^{2} + k x-1 is divided by x - 2 is 15.
The remainder when 3 x^{3} + 2 x^{2} + x + k is divided by x + 1 is -3.
The remainder when 2 x^{3} + 2 x^{2} + kx -2 is divided by x + 3 is -80.
The remainder when 2 x^{3} - 2 x^{2} - 4x + k is divided by x + 3 is -57.
The remainder when 4 x^{3} + 3 x^{2} + kx -2 is divided by x + 1 is -147.
For each of the following problems:
Perform the long division and identify the divisor, dividend, quotient, and remainder.
State whether the divisor is a factor of the dividend.
Write the dividend as a multiple of the divisor, plus a remainder if applicable.
For each of the following problems:
Perform synthetic division and identify the divisor, dividend, quotient, and remainder.
State whether the divisor is a factor of the dividend.
Write the dividend as a multiple of the divisor, plus a remainder if applicable.
The function h \left(x\right) is shown on the graph and has a leading coefficient of -1.
State the coordinates of the y-intercept of h \left(x\right).
State the zero(s) of h \left(x\right).
Write h \left(x\right) in factored form.
The function d \left(x\right) is shown on the graph, has a leading coefficient of 1 and a degree of 3.
State the zero(s) of d \left(x\right).
Determine the multiplicity of each zero.
Write d \left(x\right) in factored form.
Write the equation for d \left(x\right) in standard form.
For each of the following divisions:
\dfrac{x^3-1}{x+1}
\dfrac{x^3-x^2-32x+60}{x-3}
\dfrac{x^3+4x^2+x-6}{x+2}
\dfrac{2x^3+x^2-2x-1}{x-2}
Consider the polynomial P\left(x\right)=25x^5-75x^4-x+3.
Write down all the possible rational zeros.
Find the value of P \left( - 1 \right).
Find the value of P \left( 3 \right).
The Factor Theorem states that if P\left(a\right)=0, then x-a is a factor of P\left(x\right).
Since P\left(3\right)=0 then \left(x-3\right) is a factor of P\left(x\right). Use long division to find the quotient when P\left(x\right) is divided by x-3.
Use the substitution m=5x^2 to factorise the polynomial 25x^4-1.
Now consider the factorised polynomial \left(5x^2+1\right)\left(5x^2-1\right)=0.
Each expression is a quadratic, but only \left(5x^2-1\right) has a discriminant greater than or equal to zero, and hence real zeros.
Rearrange the equation \left(5x^2-1\right)=0 to solve for x.
Factorise P\left(x\right)=25x^5-75x^4-x+3.
Consider the polynomial P \left( x \right) = x^{3} + 4 x^{2} + x - 6
Write down all the possible rational zeros.
The graph of P \left( x \right) is shown.
State which of the possible zeros listed in the previous part are actual zeros of P \left( x \right).
Factorise P \left( x \right).
The amount of dust, in milligrams, collected on a table after x days without dusting is modeled by the function f \left( x \right) = 9 x^{3} - 30 x^{2} - 34 x + 60. After how many days without dusting will the table have collected 20 \text{ mg} of dust?
Consider the equation f(x) = g(x)q(x) + r(x). Given that f(x) = x^4 - x^3 + 2x^2 - x + 1, g(x) = x^2 - 1, and r(x) = 2x + 3, find q(x).
It costs \left(x^5-3x^4-33x^3-32x^2+26x-4 \right) dollars to replace the lawn in the backyard. If the new lawn costs \left(x + 2\right) dollars per square foot:
Find the area of the lawn in square feet.
If the lawn is rectangular and the width of the lawn is \left(x^2-8x+2\right), draw a diagram of the lawn with all sides labeled.
Consider \left( 2 x^{3} + 8 x^{2} + x + 4\right) \div \left(x + 4\right).
Show that x + 4 is a factor of the polynomial P \left( x \right) = 2 x^{3} + 8 x^{2} + x + 4 .
By completing the gaps, determine the other quadratic factor.
2 x^{3} + 8 x^{2} + x + 4 = \left(x + 4\right) \left( ⬚ x^{2} + ⬚\right)
The polynomials 4 x^{2} - 7 x - 15 and 5 x^{2} + 13 x + k have a common factor of x + p, where p is an integer.
Solve for the value of p.
Solve for the value of k.
Consider P \left( x \right) = 6 x^{3} - 11 x^{2} + 6 x-1 and the table of values for P \left( x \right) on the integers in the interval 0 \leq x \leq 5 as shown:
| x | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| P(x) | -1 | 0 | 15 | 80 | 231 | 504 |
Use the table to find a solution to the equation 6 x^{3} - 11 x^{2} + 6 x-1 = 0.
Use synthetic division to divide P \left( x \right) by x-1.
Determine whether 1 is a zero of P \left( x \right).
Now, find all the solutions of the equation 6 x^{3} - 11 x^{2} + 6 x-1 = 0.
Consider P \left( x \right) = x^{3} + 3 x^{2} - x - 3 and the graph of y = P \left( x \right) shown:
Use the graph to find a solution to the equation x^{3} + 3 x^{2} - x - 3 = 0.
Use synthetic division to divide P(x) by x -1.
Determine whether 1 is a root of P \left( x \right).
Factor x^{3} + 3 x^{2} - x - 3 completely.
Now, find all the solutions of the equation x^{3} + 3 x^{2} - x - 3 = 0.