Elementary Algebra

# Review Exercises

Elementary AlgebraReview Exercises

### Review Exercises

##### 7.1 Greatest Common Factor and Factor by Grouping

Find the Greatest Common Factor of Two or More Expressions

In the following exercises, find the greatest common factor.

363.

$42 , 60 42 , 60$

364.

$450 , 420 450 , 420$

365.

$90 , 150 , 105 90 , 150 , 105$

366.

$60 , 294 , 630 60 , 294 , 630$

Factor the Greatest Common Factor from a Polynomial

In the following exercises, factor the greatest common factor from each polynomial.

367.

$24 x − 42 24 x − 42$

368.

$35 y + 84 35 y + 84$

369.

$15 m 4 + 6 m 2 n 15 m 4 + 6 m 2 n$

370.

$24 p t 4 + 16 t 7 24 p t 4 + 16 t 7$

Factor by Grouping

In the following exercises, factor by grouping.

371.

$a x − a y + b x − b y a x − a y + b x − b y$

372.

$x 2 y − x y 2 + 2 x − 2 y x 2 y − x y 2 + 2 x − 2 y$

373.

$x 2 + 7 x − 3 x − 21 x 2 + 7 x − 3 x − 21$

374.

$4 x 2 − 16 x + 3 x − 12 4 x 2 − 16 x + 3 x − 12$

375.

$m 3 + m 2 + m + 1 m 3 + m 2 + m + 1$

376.

$5 x − 5 y − y + x 5 x − 5 y − y + x$

##### 7.2 Factor Trinomials of the form $x2+bx+cx2+bx+c$

Factor Trinomials of the Form $x2+bx+cx2+bx+c$

In the following exercises, factor each trinomial of the form $x2+bx+cx2+bx+c$.

377.

$u 2 + 17 u + 72 u 2 + 17 u + 72$

378.

$a 2 + 14 a + 33 a 2 + 14 a + 33$

379.

$k 2 − 16 k + 60 k 2 − 16 k + 60$

380.

$r 2 − 11 r + 28 r 2 − 11 r + 28$

381.

$y 2 + 6 y − 7 y 2 + 6 y − 7$

382.

$m 2 + 3 m − 54 m 2 + 3 m − 54$

383.

$s 2 − 2 s − 8 s 2 − 2 s − 8$

384.

$x 2 − 3 x − 10 x 2 − 3 x − 10$

Factor Trinomials of the Form $x2+bxy+cy2x2+bxy+cy2$

In the following examples, factor each trinomial of the form $x2+bxy+cy2x2+bxy+cy2$.

385.

$x 2 + 12 x y + 35 y 2 x 2 + 12 x y + 35 y 2$

386.

$u 2 + 14 u v + 48 v 2 u 2 + 14 u v + 48 v 2$

387.

$a 2 + 4 a b − 21 b 2 a 2 + 4 a b − 21 b 2$

388.

$p 2 − 5 p q − 36 q 2 p 2 − 5 p q − 36 q 2$

##### 7.3 Factoring Trinomials of the form $ax2+bx+cax2+bx+c$

Recognize a Preliminary Strategy to Factor Polynomials Completely

In the following exercises, identify the best method to use to factor each polynomial.

389.

$y 2 − 17 y + 42 y 2 − 17 y + 42$

390.

$12 r 2 + 32 r + 5 12 r 2 + 32 r + 5$

391.

$8 a 3 + 72 a 8 a 3 + 72 a$

392.

$4 m − m n − 3 n + 12 4 m − m n − 3 n + 12$

Factor Trinomials of the Form $ax2+bx+cax2+bx+c$ with a GCF

In the following exercises, factor completely.

393.

$6 x 2 + 42 x + 60 6 x 2 + 42 x + 60$

394.

$8 a 2 + 32 a + 24 8 a 2 + 32 a + 24$

395.

$3 n 4 − 12 n 3 − 96 n 2 3 n 4 − 12 n 3 − 96 n 2$

396.

$5 y 4 + 25 y 2 − 70 y 5 y 4 + 25 y 2 − 70 y$

Factor Trinomials Using the “ac” Method

In the following exercises, factor.

397.

$2 x 2 + 9 x + 4 2 x 2 + 9 x + 4$

398.

$3 y 2 + 17 y + 10 3 y 2 + 17 y + 10$

399.

$18 a 2 − 9 a + 1 18 a 2 − 9 a + 1$

400.

$8 u 2 − 14 u + 3 8 u 2 − 14 u + 3$

401.

$15 p 2 + 2 p − 8 15 p 2 + 2 p − 8$

402.

$15 x 2 + 6 x − 2 15 x 2 + 6 x − 2$

403.

$40 s 2 − s − 6 40 s 2 − s − 6$

404.

$20 n 2 − 7 n − 3 20 n 2 − 7 n − 3$

Factor Trinomials with a GCF Using the “ac” Method

In the following exercises, factor.

405.

$3 x 2 + 3 x − 36 3 x 2 + 3 x − 36$

406.

$4 x 2 + 4 x − 8 4 x 2 + 4 x − 8$

407.

$60 y 2 − 85 y − 25 60 y 2 − 85 y − 25$

408.

$18 a 2 − 57 a − 21 18 a 2 − 57 a − 21$

##### 7.4 Factoring Special Products

Factor Perfect Square Trinomials

In the following exercises, factor.

409.

$25 x 2 + 30 x + 9 25 x 2 + 30 x + 9$

410.

$16 y 2 + 72 y + 81 16 y 2 + 72 y + 81$

411.

$36 a 2 − 84 a b + 49 b 2 36 a 2 − 84 a b + 49 b 2$

412.

$64 r 2 − 176 r s + 121 s 2 64 r 2 − 176 r s + 121 s 2$

413.

$40 x 2 + 360 x + 810 40 x 2 + 360 x + 810$

414.

$75 u 2 + 180 u + 108 75 u 2 + 180 u + 108$

415.

$2 y 3 − 16 y 2 + 32 y 2 y 3 − 16 y 2 + 32 y$

416.

$5 k 3 − 70 k 2 + 245 k 5 k 3 − 70 k 2 + 245 k$

Factor Differences of Squares

In the following exercises, factor.

417.

$81 r 2 − 25 81 r 2 − 25$

418.

$49 a 2 − 144 49 a 2 − 144$

419.

$169 m 2 − n 2 169 m 2 − n 2$

420.

$64 x 2 − y 2 64 x 2 − y 2$

421.

$25 p 2 − 1 25 p 2 − 1$

422.

$1 − 16 s 2 1 − 16 s 2$

423.

$9 − 121 y 2 9 − 121 y 2$

424.

$100 k 2 − 81 100 k 2 − 81$

425.

$20 x 2 − 125 20 x 2 − 125$

426.

$18 y 2 − 98 18 y 2 − 98$

427.

$49 u 3 − 9 u 49 u 3 − 9 u$

428.

$169 n 3 − n 169 n 3 − n$

Factor Sums and Differences of Cubes

In the following exercises, factor.

429.

$a 3 − 125 a 3 − 125$

430.

$b 3 − 216 b 3 − 216$

431.

$2 m 3 + 54 2 m 3 + 54$

432.

$81 x 3 + 3 81 x 3 + 3$

##### 7.5 General Strategy for Factoring Polynomials

Recognize and Use the Appropriate Method to Factor a Polynomial Completely

In the following exercises, factor completely.

433.

$24 x 3 + 44 x 2 24 x 3 + 44 x 2$

434.

$24 a 4 − 9 a 3 24 a 4 − 9 a 3$

435.

$16 n 2 − 56 m n + 49 m 2 16 n 2 − 56 m n + 49 m 2$

436.

$6 a 2 − 25 a − 9 6 a 2 − 25 a − 9$

437.

$5 r 2 + 22 r − 48 5 r 2 + 22 r − 48$

438.

$5 u 4 − 45 u 2 5 u 4 − 45 u 2$

439.

$n 4 − 81 n 4 − 81$

440.

$64 j 2 + 225 64 j 2 + 225$

441.

$5 x 2 + 5 x − 60 5 x 2 + 5 x − 60$

442.

$b 3 − 64 b 3 − 64$

443.

$m 3 + 125 m 3 + 125$

444.

$2 b 2 − 2 b c + 5 c b − 5 c 2 2 b 2 − 2 b c + 5 c b − 5 c 2$

Use the Zero Product Property

In the following exercises, solve.

445.

$( a − 3 ) ( a + 7 ) = 0 ( a − 3 ) ( a + 7 ) = 0$

446.

$( b − 3 ) ( b + 10 ) = 0 ( b − 3 ) ( b + 10 ) = 0$

447.

$3 m ( 2 m − 5 ) ( m + 6 ) = 0 3 m ( 2 m − 5 ) ( m + 6 ) = 0$

448.

$7 n ( 3 n + 8 ) ( n − 5 ) = 0 7 n ( 3 n + 8 ) ( n − 5 ) = 0$

In the following exercises, solve.

449.

$x 2 + 9 x + 20 = 0 x 2 + 9 x + 20 = 0$

450.

$y 2 − y − 72 = 0 y 2 − y − 72 = 0$

451.

$2 p 2 − 11 p = 40 2 p 2 − 11 p = 40$

452.

$q 3 + 3 q 2 + 2 q = 0 q 3 + 3 q 2 + 2 q = 0$

453.

$144 m 2 − 25 = 0 144 m 2 − 25 = 0$

454.

$4 n 2 = 36 4 n 2 = 36$

Solve Applications Modeled by Quadratic Equations

In the following exercises, solve.

455.

The product of two consecutive numbers is $462462$. Find the numbers.

456.

The area of a rectangular shaped patio $400400$ square feet. The length of the patio is $99$ feet more than its width. Find the length and width.

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