The question asks which expression is a factor of $x^3 - 4x^2 + 4x$. First, factor out the common factor of $x$: $x^3 - 4x^2 + 4x = x(x^2 - 4x + 4) = x(x-2)^2$. The complete factorization shows that the factors are $x$, $(x-2)$, and $(x-2)$ again. Among the choices, only $(x)$ appears as a factor. A common error is not recognizing the need to factor out the greatest common factor first. When looking for factors, always start by factoring out any common terms before attempting other factoring methods.

The question asks for the zeros of $x^2 - 5x + 6$. To find zeros, we set the polynomial equal to zero and solve: $x^2 - 5x + 6 = 0$. This factors as $(x-2)(x-3) = 0$, giving us $x = 2$ or $x = 3$. We can verify by substitution: $2^2 - 5(2) + 6 = 4 - 10 + 6 = 0$ ✓ and $3^2 - 5(3) + 6 = 9 - 15 + 6 = 0$ ✓. A common error is confusing zeros with factors or making sign errors when factoring. Always verify your zeros by substituting back into the original equation.

The question asks for $f(2)$ when $f(x) = x^2 - 4x + 4$. Substitute $x = 2$: $f(2) = (2)^2 - 4(2) + 4 = 4 - 8 + 4 = 0$. Notice that this polynomial can be factored as $(x-2)^2$, which confirms that $x = 2$ is a zero (making $f(2) = 0$). A common error is making arithmetic mistakes during substitution or incorrectly handling negative signs. When evaluating polynomial functions, substitute carefully and double-check your arithmetic, especially with multiple operations.

The question asks for the product of $(x + 2)$ and $(x - 3)$. Using the FOIL method: First terms give $x \cdot x = x^2$, Outer terms give $x \cdot(-3) = -3x$, Inner terms give $2 \cdot x = 2x$, and Last terms give $2 \cdot(-3) = -6$. Combining these: $x^2 - 3x + 2x - 6 = x^2 - x - 6$. A common error is incorrectly combining the middle terms or making sign errors during multiplication. Always double-check your arithmetic when expanding polynomial products, especially with negative coefficients.

The question asks for the number of x-intercepts of $x^2 + 4x + 5$. X-intercepts occur where the polynomial equals zero, so solve $x^2 + 4x + 5 = 0$. Using the discriminant: $b^2 - 4ac = 4^2 - 4(1)(5) = 16 - 20 = -4$. Since the discriminant is negative, there are no real solutions, meaning no x-intercepts. This parabola opens upward (positive leading coefficient) and lies entirely above the x-axis. A common error is attempting to factor without checking if real solutions exist. For quadratics, the discriminant quickly determines the number of real roots and x-intercepts.

The question asks for the factored form of $x^2 - 9$. This is a difference of squares pattern: $a^2 - b^2 = (a-b)(a+b)$. Here, $x^2 - 9 = x^2 - 3^2 = (x-3)(x+3)$. We can verify: $(x-3)(x+3) = x^2 + 3x - 3x - 9 = x^2 - 9$ ✓. A common error is confusing this with perfect square trinomials like $(x-3)^2$ or $(x+3)^2$. Remember that difference of squares always factors into two binomials with opposite signs.

This question asks for the zeros (roots) of a quadratic polynomial, which are the x-values where the polynomial equals zero. To find the zeros of x² - 4x + 3, we can factor it by finding two numbers that multiply to 3 and add to -4, which are -1 and -3. This gives us (x - 1)(x - 3) = 0, so x = 1 or x = 3.

This question asks for the number of x-intercepts of a cubic polynomial, which corresponds to the number of real zeros. For x³ - 3x² + 2x, we first factor out the common factor x to get x(x² - 3x + 2). Then we factor the quadratic: x² - 3x + 2 = (x - 1)(x - 2), giving us the complete factorization x(x - 1)(x - 2). Setting each factor equal to zero gives us x = 0, x = 1, and x = 2, which means there are 3 x-intercepts. A key insight is that factoring reveals all real zeros directly, and each distinct zero corresponds to one x-intercept. Always factor completely to identify all intercepts of a polynomial graph.

This question requires factoring a quadratic polynomial of the form x² + bx + c. For x² - 5x + 6, we need two numbers that multiply to 6 and add to -5. The factors of 6 are 1×6 and 2×3, and since we need a sum of -5, we use -2 and -3 because (-2) + (-3) = -5 and (-2)(-3) = 6. This gives us the factored form (x - 2)(x - 3). A common error is mixing up the signs or finding factors that multiply correctly but don't add to the middle coefficient. Always verify your factorization by expanding it back to the original polynomial.

This question asks us to identify the degree of a polynomial, which is the highest power of the variable. In the polynomial 4x³ - 2x² + 7x - 5, we examine each term's exponent: the x³ term has degree 3, the x² term has degree 2, the x term has degree 1, and the constant term has degree 0. The degree of the entire polynomial is the highest of these individual degrees, which is 3. A common mistake is confusing the degree with the number of terms or the leading coefficient. To find polynomial degree, always look for the term with the largest exponent on the variable.