Chapter 6

Solve the system by graphing: \(\left\\{\begin{array}{rr}2 x-y= & -4 \\ x-3 y= & 3\end{array}\right.\) (Section 4.1, Example 2)

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The polynomial \(16 x^{2}+20 x+25\) is a perfect square trinomial.

A rock is dropped from the top of a 256 -foot cliff. The height, in feet, of the rock above the water after \(t\) seconds is modeled by the polynomial \(256-16 t^{2} .\) Factor this expression completely. (Image can't copy)

Write the point-slope form of the equation of the line passing through \((-7,2)\) and \((-4,5) .\) Then use the point-slope form of the equation to write the slope-intercept form of the equation. (Section 3.5 Example 2 )

Will help you prepare for the material covered in the next section. Find two factors of 8 whose sum is 6

Factor each polynomial. $$x^{2}-y^{2}+3 x+3 y$$

Factor each polynomial. $$x^{2 n}-25 y^{2 n}$$

Describe a strategy that can be used to factor polynomials.

Will help you prepare for the material covered in the next section. Find two factors of \(-35\) whose sum is \(2 .\)

Factor each polynomial. $$4 x^{2 n}+12 x^{n}+9$$

Describe some of the difficulties in factoring polynomials. What suggestions can you offer to overcome these difficulties?

Factor each polynomial. $$(x+3)^{2}-2(x+3)+1$$

You are about to take a great picture of fog rolling into San Francisco from the middle of the Golden Gate Bridge, 400 feet above the water. Whoops! You accidently lean too far over the safety rail and drop your camera. The height, in feet, of the camera after \(t\) seconds is modeled by the polynomial \(400-16 t^{2} .\) The factored form of the polynomial is \(16(5+t)(5-t) .\) Describe something about your falling camera that is easier to see from the factored form, \(16(5+t)(5-t),\) than from the form \(400-16 t^{2}\)

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. It takes a great deal of practice to get good at factoring a wide variety of polynomials.

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. Multiplying polynomials is relatively mechanical, but factoring often requires a great deal of thought.

Use the \([\mathrm{GRAPH}]\) or \([\mathrm{TABLE}]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If the graphs of \(y_{1}\) and \(y_{2}\) coincide, or if their corresponding table values are equal, this means that the polynomial on the left side has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$4 x^{2}-9=(4 x+3)(4 x-3)$$

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. The factorable trinomial \(4 x^{2}+8 x+3\) and the prime trinomial \(4 x^{2}+8 x+1\) are in the form \(a x^{2}+b x+c\) but \(b^{2}-4 a c\) is a perfect square only in the case of the factorable trinomial.

Use the \([\mathrm{GRAPH}]\) or \([\mathrm{TABLE}]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If the graphs of \(y_{1}\) and \(y_{2}\) coincide, or if their corresponding table values are equal, this means that the polynomial on the left side has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$x^{2}-6 x+9=(x-3)^{2}$$

Determine whether each statement “makes sense” or “does not make sense” and explain your reasoning. When a factorization requires two factoring techniques, I'm less likely to make errors if I show one technique at a time rather than combining the two factorizations into one step.

Use the \([\mathrm{GRAPH}]\) or \([\mathrm{TABLE}]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If the graphs of \(y_{1}\) and \(y_{2}\) coincide, or if their corresponding table values are equal, this means that the polynomial on the left side has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$4 x^{2}-4 x+1=(4 x-1)^{2}$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \(x^{2}-9=(x-3)^{2}\) for any real number \(x\)

Use the \([\mathrm{GRAPH}]\) or \([\mathrm{TABLE}]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If the graphs of \(y_{1}\) and \(y_{2}\) coincide, or if their corresponding table values are equal, this means that the polynomial on the left side has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$x^{3}-1=(x-1)\left(x^{2}-x+1\right)$$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The polynomial \(4 x^{2}+100\) is the sum of two squares and therefore cannot be factored.

$$\text { Simplify: }\left(2 x^{2} y^{3}\right)^{4}\left(5 x y^{2}\right)$$

The polynomial \(4 x^{2}+100\) is the sum of two squares and therefore cannot be factored. If the general factoring strategy is used to factor a polynomial, at least two factorizations are necessary before the given polynomial is factored completely.

$$\text { Subtract: }\left(10 x^{2}-5 x+2\right)-\left(14 x^{2}-5 x-1\right)$$

$$\text { Divide: } \frac{6 x^{2}+11 x-10}{3 x-2}$$

Factor completely. $$3 x^{5}-21 x^{3}-54 x$$

Factor completely. $$3 x^{3}-75 x$$

Factor completely. $$5 y^{5}-5 y^{4}-20 y^{3}+20 y^{2}$$

Factor completely. $$2 x^{2}-20 x+50$$

Factor completely. $$4 x^{4}-9 x^{2}+5$$

Factor completely. $$x^{3}-2 x^{2}-x+2$$

Factor completely. $$(x+5)^{2}-20(x+5)+100$$

Factor completely. $$3 x^{2 n}-27 y^{2 n}$$

Use the \([\mathrm{GRAPH}]\) or \([\mathrm { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$4 x^{2}-12 x+9=(4 x-3)^{2} ;[-5,5,1] \text { by }[0,20,1]$$

Use the \([\mathrm{GRAPH}]\) or \([\mathrm { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$\begin{aligned}&3 x^{3}-12 x^{2}-15 x=3 x(x+5)(x-1) ;[-5,7,1] \text { by }\\\ &[-80,80,10] \end{aligned}$$

Use the \([\mathrm{GRAPH}]\) or \([\mathrm { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$\begin{aligned} &6 x^{2}+10 x-4=2(3 x-1)(x+2) ;[-5,5,1] \text { by }\\\ &[-20,20,2] \end{aligned}$$

Use the \([\mathrm{GRAPH}]\) or \([\mathrm { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$\begin{aligned} &x^{4}-16=\left(x^{2}+4\right)(x+2)(x-2) ;[-5,5,1] \text { by }\\\ &[-20,20,2] \end{aligned}$$

Use the \([\mathrm{GRAPH}]\) or \([\mathrm { TABLE }]\) feature of a graphing utility to determine if the polynomial on the left side of each equation has been correctly factored. If not, factor the polynomial correctly and then use your graphing utility to verify the factorization. $$\begin{aligned} &-2 x^{3}+10 x^{2}-2 x-10=2(x+5)\left(x^{2}+1\right) ;[-8,4,1] \text { by }\\\ &[-100,100,10] \end{aligned}$$

Factor: \(9 x^{2}-16 .\)

Graph using intercepts: \(5 x-2 y=10\)

The second angle of a triangle measures three times that of the first angle's measure. The third angle measures \(80^{\circ}\) more than the first. Find the measure of each angle.

Exercises 150–152 will help you prepare for the material covered in the next section. Evaluate \((3 x-1)(x+2)\) for \(x=\frac{1}{3}\)

Exercises 150–152 will help you prepare for the material covered in the next section. Evaluate \(2 x^{2}+7 x-4\) for \(x=\frac{1}{2}\)

Exercises 150–152 will help you prepare for the material covered in the next section. Factor: \((x-2)(x+3)-6\)