Question

1. the quadratic expression ( x^2 - 8x + 10 ) has its smallest value for some integer value of ( x ) on the interval ( 0 leq x leq 10 ). set up a table to find the smallest value of the expression and the value of ( x ) that gives this value. show your table below.

1. consider the complex expression ( (x + 7)(x + 3) + (x - 1)(x - 4) ).

(a) multiply the two sets of binomials and combine like terms in order to write this expression as an equivalent trinomial in standard form. show your work.

(b) set up a table to verify that your answer in part (a) is equivalent to the original expression. dont hesitate to point out that it is not equivalent (which means you either made a mistake in your algebra or in your table set up). show your table.

1. the product of three binomials is shown below. write this product as a polynomial in standard form. (its highest power will be ( x^3 )).

( (x - 1)(x + 2)(x - 4) )

1. set up a table for the answer you found in #6 on the interval ( -5 leq x leq 5 ). where does this expression have zeroes?

Explanation:

Problem 4

Step1: Create table for integer x

$x$	$x^2 - 8x + 10$
1	$1^2 -8(1)+10=3$
2	$2^2 -8(2)+10=-2$
3	$3^2 -8(3)+10=-5$
4	$4^2 -8(4)+10=-6$
5	$5^2 -8(5)+10=-5$
6	$6^2 -8(6)+10=-2$
7	$7^2 -8(7)+10=3$
8	$8^2 -8(8)+10=10$
9	$9^2 -8(9)+10=19$
10	$10^2 -8(10)+10=30$

Step2: Identify minimum value

Scan the calculated values to find the smallest.

Step1: Multiply first binomial pair

$(x+7)(x+3) = x^2 +3x +7x +21 = x^2 +10x +21$

Step2: Multiply second binomial pair

$(x-1)(x-4) = x^2 -4x -x +4 = x^2 -5x +4$

Step3: Add and combine like terms

$(x^2 +10x +21) + (x^2 -5x +4) = 2x^2 +5x +25$

Step1: Choose test x values

Pick integer values for $x$ to test both expressions.

Step2: Build verification table

$x$	Original Expression $(x+7)(x+3)+(x-1)(x-4)$	Simplified Expression $2x^2+5x+25$
-1	$(6)(2)+(-2)(-5)=12+10=22$	$2(1)+5(-1)+25=2-5+25=22$
0	$(7)(3)+(-1)(-4)=21+4=25$	$2(0)+5(0)+25=25$
1	$(8)(4)+(0)(-3)=32+0=32$	$2(1)+5(1)+25=2+5+25=32$
2	$(9)(5)+(1)(-2)=45-2=43$	$2(4)+5(2)+25=8+10+25=43$

Step3: Confirm equivalence

All test values produce matching results, so the expressions are equivalent.

Answer:

The smallest value is $-6$, which occurs when $x=4$.

---

Problem 5 (a)