Question

module 6.2: exit ticket

1. perform the following polynomial multiplications.

$(x^2 + 9x + 7)(3x^2 + 9x + 5)$

1. write a polynomial function to represent the new value

the volume of a stock, or number of shares traded, is modeled over time during a given day by $s(x) = x^5 - 3x^4 + 10x^2 - 6x + 30$. the cost per share of that stock during that day is modeled by $c(x) = 0.004x^4 - 0.02x^2 + 0.3x + 4$. write a polynomial function $v(x)$ to model the changing value during that day of the trades made of shares of that stock.

1. verify the given polynomial identity

$x^4 - y^4 = (x - y)(x + y)(x^2 + y^2)$

Explanation:

Problem 1:

Step1: Apply distributive property (FOIL for polynomials)

Multiply each term in the first polynomial by each term in the second polynomial:

$$
LATEXBLOCK0
$$

Step2: Simplify each product

$$
LATEXBLOCK1
$$

Step3: Combine like terms

Combine the terms with the same power of \(x\):

• \(x^4\): \(3x^4\)
• \(x^3\): \(9x^3 + 27x^3 = 36x^3\)
• \(x^2\): \(5x^2 + 81x^2 + 21x^2 = 107x^2\)
• \(x\): \(45x + 63x = 108x\)
• Constant: \(35\)

So the result is \(3x^4 + 36x^3 + 107x^2 + 108x + 35\)

Step1: Recall the formula for value

The value \(V(x)\) of the trades is the product of the volume \(S(x)\) and the cost per share \(C(x)\), so \(V(x)=S(x)\times C(x)\)

Step2: Multiply the polynomials

$$
LATEXBLOCK0
$$

Step3: Simplify each product

$$
LATEXBLOCK1
$$

Step4: Combine like terms

• \(x^9\): \(0.004x^9\)
• \(x^8\): \(-0.012x^8\)
• \(x^7\): \(-0.02x^7\)
• \(x^6\): \(0.3x^6 + 0.06x^6 + 0.04x^6 = 0.4x^6\)
• \(x^5\): \(4x^5 - 0.9x^5 - 0.024x^5 = 3.076x^5\)
• \(x^4\): \(-12x^4 - 0.2x^4 + 0.12x^4 = -12.08x^4\)
• \(x^3\): \(3x^3 + 0.12x^3 = 3.12x^3\)
• \(x^2\): \(40x^2 - 1.8x^2 - 0.6x^2 = 37.6x^2\)
• \(x\): \(-24x + 9x = -15x\)
• Constant: \(120\)

So \(V(x)=0.004x^9 - 0.012x^8 - 0.02x^7 + 0.4x^6 + 3.076x^5 - 12.08x^4 + 3.12x^3 + 37.6x^2 - 15x + 120\)

Step1: Start with the right-hand side (RHS)

First, multiply \((x - y)(x + y)\) using the difference of squares formula \((a - b)(a + b)=a^2 - b^2\), so \((x - y)(x + y)=x^2 - y^2\)

Step2: Multiply the result by \((x^2 + y^2)\)

Now multiply \((x^2 - y^2)(x^2 + y^2)\) again using the difference of squares formula, where \(a = x^2\) and \(b = y^2\). So \((x^2 - y^2)(x^2 + y^2)=(x^2)^2 - (y^2)^2=x^4 - y^4\)

Step3: Compare with left-hand side (LHS)

The left-hand side is \(x^4 - y^4\), which is equal to the result from the right-hand side. So the identity is verified.

Answer:

\(3x^4 + 36x^3 + 107x^2 + 108x + 35\)

Problem 2: