Question

notes: ★ roots / zeros / x-intercept: the point(s) where the graph intersects the x-axis. all three words mean the same thing! type the equations into desmos and see where the graph touches the x-axis (side to side) model: 1. which function has the zeros -1, 3, and -4? 1) $f(x) = (x + 1)(x - 3)(x - 4)$ 2) $g(x) = (x - 1)(x + 3)(x - 4)$ 3) $h(x) = (x + 1)(x - 3)(x + 4)$ 4) $k(x) = (x - 1)(x + 3)(x + 4)$ 2. which polynomial function has zeros at -3, 0, a 1) $f(x) = (x + 3)(x^2 + 4)$ 2) $f(x) = (x^2 - 3)(x - 4)$ 3) $f(x) = x(x + 3)(x - 4)$ 4) $f(x) = x(x - 3)(x + 4)$ 3. the zeros of the function $f(x) = 3x^2 - 3x - 6$ are 1) -1 and -2 2) 1 and -2 3) 1 and 2 4) -1 and 2 4. keith determines the zeros of the function f(x) -6 and 5. what could be keith’s function? 1) $f(x) = (x + 5)(x + 6)$ 2) $f(x) = (x + 5)(x - 6)$ 3) $f(x) = (x - 5)(x + 6)$ 4) $f(x) = (x - 5)(x - 6)$ 5. what are the zeros of $f(x) = x^2 - 8x - 20$? 1) 10 and 2 2) 10 and -2 3) -10 and 2 4) -10 and -2 6. what are the zeros of $m(x) = x(x^2 - 16)$? 1) -4 and 4, only 2) -8 and 8, only 3) -4, 0, and 4 4) -8, 0, and 8 7. determine all the zeros of $m(x) = x^2 - 4x + 3$ algebraically. 8. find the zeros of $f(x) = (x - 3)^2 - 49$, algebraically.

Explanation:

Let's solve question 7: Determine all the zeros of \( m(x) = x^2 - 4x + 3 \) algebraically.

Step 1: Factor the quadratic

To find the zeros, we first factor the quadratic expression \( x^2 - 4x + 3 \). We need two numbers that multiply to \( 3 \) and add up to \( -4 \). Those numbers are \( -1 \) and \( -3 \). So, we can factor it as:
\( x^2 - 4x + 3 = (x - 1)(x - 3) \)

Step 2: Set each factor to zero

To find the zeros, we set \( m(x) = 0 \), so:
\( (x - 1)(x - 3) = 0 \)
Using the zero - product property (if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \)), we set each factor equal to zero:
\( x - 1 = 0 \) or \( x - 3 = 0 \)

Step 3: Solve for x

For \( x - 1 = 0 \), adding \( 1 \) to both sides gives \( x = 1 \).
For \( x - 3 = 0 \), adding \( 3 \) to both sides gives \( x = 3 \).

Answer:

The zeros of \( m(x)=x^{2}-4x + 3\) are \( x = 1 \) and \( x = 3 \).