6.5 Polynomial Equations

Before you get started, take this readiness quiz.

Solve: $5 y - 3 = 0 .$
If you missed this problem, review the reference.

Factor completely: $n^{3} - 9 n^{2} - 22 n .$
If you missed this problem, review the reference.

If $f \left(x\right) = 8 x - 16 ,$ find $f \left(3\right)$ and solve $f \left(x\right) = 0 .$
If you missed this problem, review the reference.

Solve: $\left(3 m - 2\right) \left(2 m + 1\right) = 0 .$

Solve: $\left(4 p + 3\right) \left(4 p - 3\right) = 0 .$

Solve: $3 c^{2} = 10 c - 8 .$

Solve: $2 d^{2} - 5 d = 3 .$

Solve: $169 x^{2} = 49 .$

Solve: $25 p^{2} = 49 .$

Solve: $36 x^{2} = 121 .$

Solve: $\left(3 x - 8\right) \left(x - 1\right) = 3 x .$

Solve: $\left(2 m + 1\right) \left(m + 3\right) = 1 2 m .$

Solve: $\left(k + 1\right) \left(k - 1\right) = 8 .$

Solve: $3 x^{2} = 12 x + 63 .$

Solve: $18 a^{2} - 30 = −33 a .$

Solve: $123 b = −6 - 60 b^{2} .$

Solve: $9 m^{3} + 100 m = 60 m^{2} .$

Solve: $8 x^{3} = 24 x^{2} - 18 x .$

Solve: $16 y^{2} = 32 y^{3} + 2 y .$

For the function $f \left(x\right) = x^{2} + 2 x - 2 ,$

ⓐ find x when $f \left(x\right) = 6$ ⓑ find two points that lie on the graph of the function.

For the function $f \left(x\right) = x^{2} - 2 x - 8 ,$

ⓐ find x when $f \left(x\right) = 7$ ⓑ Find two points that lie on the graph of the function.

For the function $f \left(x\right) = x^{2} - 8 x + 3 ,$

ⓐ find x when $f \left(x\right) = −4$ ⓑ Find two points that lie on the graph of the function.

For the function $f \left(x\right) = 3 x^{2} + 10 x - 8 ,$ find

ⓐ the zeros of the function, ⓑ any x-intercepts of the graph of the function, ⓒ any y-intercepts of the graph of the function

For the function $f \left(x\right) = 2 x^{2} - 7 x + 5 ,$ find

ⓐ the zeros of the function, ⓑ any x-intercepts of the graph of the function, ⓒ any y-intercepts of the graph of the function.

For the function $f \left(x\right) = 6 x^{2} + 13 x - 15 ,$ find

ⓐ the zeros of the function, ⓑ any x-intercepts of the graph of the function, ⓒ any y-intercepts of the graph of the function.

The product of two consecutive odd integers is 323. Find the integers.

The product of two consecutive odd integers is 255. Find the integers.

The product of two consecutive odd integers is 483 Find the integers.

A rectangular sign has an area of 30 square feet. The length of the sign is one foot more than the width. Find the length and width of the sign.

A rectangular patio has an area of 180 square feet. The width of the patio is three feet less than the length. Find the length and width of the patio.

Justine wants to put a deck in the corner of her backyard in the shape of a right triangle. The length of one side of the deck is 7 feet more than the other side. The hypotenuse is 13. Find the lengths of the two sides of the deck.

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of the other leg. Find the lengths of the hypotenuse and the other leg.

Dennis is going to throw his rubber band ball upward from the top of a campus building. When he throws the rubber band ball from 80 feet above the ground, the function $h \left(t\right) = −16 t^{2} + 64 t + 80$ models the height, h, of the ball above the ground as a function of time, t. Find:

ⓐ the zeros of this function which tell us when the ball hits the ground, ⓑ when the ball will be 80 feet above the ground, ⓒ the height of the bal…

Genevieve is going to throw a rock from the top a trail overlooking the ocean. When she throws the rock upward from 160 feet above the ocean, the function $h \left(t\right) = −16 t^{2} + 48 t + 160$ models the height, h, of the rock above the ocean as a function of time, t. Find:

ⓐ the zeros of this function which tell us when the rock will hit the ocean, ⓑ when the rock will be 160 feet above the ocean, ⓒ the height of the rock at $t = 1.5$ seco…

Calib is going to throw his lucky penny from his balcony on a cruise ship. When he throws the penny upward from 128 feet above the ground, the function $h \left(t\right) = −16 t^{2} + 32 t + 128$ models the height, h, of the penny above the ocean as a function of time, t. Find:

ⓐ the zeros of this function which is when the penny will hit the ocean, ⓑ when the penny will be 128 feet above the ocean, ⓒ the height the penny will be at $t = 1$ sec…

Solve.

$\left(3 a - 10\right) \left(2 a - 7\right) = 0$

Solve.

$6 m \left(12 m - 5\right) = 0$

Solve.

$\left(2 x - 1\right)^{2} = 0$

Solve.

$5 a^{2} - 26 a = 24$

Solve.

$4 m^{2} = 17 m - 15$

Solve.

$7 a^{2} + 14 a = 7 a$

Solve.

$49 m^{2} = 144$

Solve.

$16 y^{2} = 81$

Solve.

$121 n^{2} = 36$

Solve.

$\left(x + 6\right) \left(x - 3\right) = −8$

Solve.

$\left(2 x + 1\right) \left(x - 3\right) = −4 x$

Solve.

$\left(3 x - 2\right) \left(x + 4\right) = 12 x$

Solve.

$20 x^{2} - 60 x = −45$

Solve.

$15 x^{2} - 10 x = 40$

Solve.

$18 x^{2} - 9 = −21 x$

Solve.

$16 p^{3} = 24 p^{2} – 9 p$

Solve.

$2 x^{3} + 72 x = 24 x^{2}$

Solve.

$36 x^{3} + 24 x^{2} = −4 x$

Solve.

For the function, $f \left(x\right) = x^{2} - 8 x + 8 ,$ ⓐ find when $f \left(x\right) = −4$ ⓑ Use this information to find two points that lie on the graph of the function.

Solve.

For the function, $f \left(x\right) = 8 x^{2} - 18 x + 5 ,$ ⓐ find when $f \left(x\right) = −4$ ⓑ Use this information to find two points that lie on the graph of the function.

For each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.

$f \left(x\right) = 9 x^{2} - 4$

For each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.

$f \left(x\right) = 6 x^{2} - 7 x - 5$

Solve.

The product of two consecutive odd integers is 143. Find the integers.

Solve.

The product of two consecutive even integers is 168. Find the integers.

Solve.

The area of a rectangular carpet is 28 square feet. The length is three feet more than the width. Find the length and the width of the carpet.

Solve.

The area of a bulletin board is 55 square feet. The length is four feet less than three times the width. Find the length and the width of the a bulletin board.

Solve.

A pennant is shaped like a right triangle, with hypotenuse 10 feet. The length of one side of the pennant is two feet longer than the length of the other side. Find the length of the two sides of the pennant.

Solve.

A reflecting pool is shaped like a right triangle, with one leg along the wall of a building. The hypotenuse is 9 feet longer than the side along the building. The third side is 7 feet longer than the side along the building. Find the lengths of all three sides of the reflecting pool.

Solve.

Juli is going to launch a model rocket in her back yard. When she launches the rocket, the function $h \left(t\right) = −16 t^{2} + 32 t$ models the height, h, of the rocket above the ground as a function of time, t. Find:

ⓐ the zeros of this function, which tell us when the rocket will be on the ground. ⓑ the time the rocket will be 16 feet above the ground.

Explain how you solve a quadratic equation. How many answers do you expect to get for a quadratic equation?

Find the greatest common factor.

$12 a^{2} b^{3} , 15 a b^{2}$

Find the greatest common factor.

$15 y^{3} , 21 y^{2} , 30 y$

Factor the greatest common factor from each polynomial.

$35 y + 84$

Factor the greatest common factor from each polynomial.

$18 x^{3} - 15 x$

Factor the greatest common factor from each polynomial.

$4 x^{3} - 12 x^{2} + 16 x$

Factor the greatest common factor from each polynomial.

$−3 x^{3} + 27 x^{2} - 12 x$

Factor by grouping.

$a x - a y + b x - b y$

Factor by grouping.

$x^{2} + 7 x - 3 x - 21$

Factor by grouping.

$m^{3} + m^{2} + m + 1$

Factor each trinomial of the form $x^{2} + b x + c .$

$a^{2} + 14 a + 33$

Factor each trinomial of the form $x^{2} + b x + c .$

$m^{2} + 3 m - 54$

In the following examples, factor each trinomial of the form $x^{2} + b x y + c y^{2} .$

$x^{2} + 12 x y + 35 y^{2}$

In the following examples, factor each trinomial of the form $x^{2} + b x y + c y^{2} .$

$a^{2} + 4 a b - 21 b^{2}$

In the following examples, factor each trinomial of the form $x^{2} + b x y + c y^{2} .$

$m^{2} - 5 m n + 30 n^{2}$

Factor completely using trial and error.

$3 y^{3} - 21 y^{2} + 30 y$

Factor completely using trial and error.

$5 y^{2} + 14 y + 9$

Factor completely using trial and error.

$10 y^{2} - 53 y - 11$

Factor completely using trial and error.

$−81 a^{2} + 153 a + 18$

Factor.

$18 a^{2} - 9 a + 1$

Factor.

$15 x^{2} + 6 x - 2$

Factor.

$3 x^{2} + 3 x - 36$

Factor.

$18 a^{2} - 57 a - 21$

Factor using substitution.

$x^{4} - 13 x^{2} - 30$

Factor completely using the perfect square trinomials pattern.

$25 x^{2} + 30 x + 9$

Factor completely using the perfect square trinomials pattern.

$40 x^{2} + 360 x + 810$

Factor completely using the perfect square trinomials pattern.

$75 u^{4} - 30 u^{3} v + 3 u^{2} v^{2}$

Factor completely using the difference of squares pattern, if possible.

$169 m^{2} - n^{2}$

Factor completely using the difference of squares pattern, if possible.

$9 - 121 y^{2}$

Factor completely using the difference of squares pattern, if possible.

$169 n^{3} - n$

Factor completely using the difference of squares pattern, if possible.

$24 p^{2} + 54$

Factor completely using the difference of squares pattern, if possible.

$16 z^{4} - 1$

Factor completely using the difference of squares pattern, if possible.

$a^{2} + 6 a + 9 - 9 b^{2}$

Factor completely using the sums and differences of cubes pattern, if possible.

$a^{3} - 125$

Factor completely using the sums and differences of cubes pattern, if possible.

$2 m^{3} + 54$

Factor completely.

$24 x^{3} + 44 x^{2}$

Factor completely.

$16 n^{2} - 56 m n + 49 m^{2}$

Factor completely.

$5 u^{4} - 45 u^{2}$

Factor completely.

$64 j^{2} + 225$

Factor completely.

$b^{3} - 64$

Factor completely.

$2 b^{2} - 2 b c + 5 c b - 5 c^{2}$

Factor completely.

$5 q^{2} - 15 q - 90$

Factor completely.

$10 m^{4} - 6250$

Factor completely.

$16 x^{2} - 24 x y + 9 y^{2} - 64$

Solve.

$\left(5 b + 1\right) \left(6 b + 1\right) = 0$

Solve.

$\left(2 x - 1\right)^{2} = 0$

Solve.

$x^{2} + 9 x + 20 = 0$

Solve.

$2 p^{2} - 11 p = 40$

Solve.

$144 m^{2} - 25 = 0$

Solve.

$\left(x + 6\right) \left(x - 3\right) = −8$

Solve.

$16 p^{3} = 24 p^{2} - 9 p$

Solve.

For the function, $f \left(x\right) = x^{2} + 11 x + 20 ,$ ⓐ find when $f \left(x\right) = −8$ ⓑ Use this information to find two points that lie on the graph of the function.

In each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.

$f \left(x\right) = 64 x^{2} - 49$

Solve.

The product of two consecutive odd numbers is 399. Find the numbers.

Solve.

A ladder leans against the wall of a building. The length of the ladder is 9 feet longer than the distance of the bottom of the ladder from the building. The distance of the top of the ladder reaches up the side of the building is 7 feet longer than the distance of the bottom of the ladder from the building. Find the lengths of all three sides of the triangle formed by the ladder leaning ag…

Factor completely.

$80 a^{2} + 120 a^{3}$

Factor completely.

$x^{2} + 13 x + 36$

Factor completely.

$x y - 8 y + 7 x - 56$

Factor completely.

$9 s^{2} - 12 s + 4$

Factor completely.

$3 x^{2} - 75 y^{2}$

Factor completely.

$x^{3} + 125$

Factor completely.

$6 x^{4} - 19 x^{2} + 15$

Solve

$5 a^{2} + 26 a = 24$

Solve

The area of a rectangular place mat is 168 square inches. Its length is two inches longer than the width. Find the length and width of the placemat.

Solve

For the function, $f \left(x\right) = x^{2} - 7 x + 5 ,$ ⓐ find when $f \left(x\right) = −7$ ⓑ Use this information to find two points that lie on the graph of the function.