Identify Terms: Identify the terms in each polynomial. In the expression (2z-1)(z^2-2z+1), (2z-1) is the first binomial and (z^2-2z+1) is the second trinomial. First Binomial Terms: 2z, -1 Second Trinomial Terms: z^2, -2z, +1Use Distributive Property: Use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first binomial by each term in the second trinomial. The distributive property states that a(b + c + d) = ab + ac + ad. Applying this to our polynomials, we get: (2z-1)(z^2) + (2z-1)(-2z) + (2z-1)(1)Distribute Terms: Distribute each term in the binomial across the terms in the trinomial. This gives us: (2z * z^2) + (2z * -2z) + (2z * 1) + (-1 * z^2) + (-1 * -2z) + (-1 * 1)Perform Multiplication: Perform the multiplication for each term. Now we multiply the coefficients and add the exponents for terms with the same base: (2z^3) + (-4z^2) + (2z) + (-z^2) + (2z) + (-1)Combine Like Terms: Combine like terms. We combine terms with the same exponents: 2z^3 + (-4z^2 - z^2) + (2z + 2z) - 1 This simplifies to: 2z^3 - 5z^2 + 4z - 1

Grade 8

Powers with negative bases

Full solution

(2z-1)(z^2-2z+1)=

Identify Terms: Identify the terms in each polynomial. $\newline$ In the expression $(2z-1)(z^2-2z+1)$ , $(2z-1)$ is the first binomial and $(z^2-2z+1)$ is the second trinomial. $\newline$ First Binomial Terms: $2z$ , $-1$ $\newline$ Second Trinomial Terms: $z^2$ , $-2z$ , $+1$
Use Distributive Property: Use the distributive property (also known as the FOIL method for binomials) to multiply each term in the first binomial by each term in the second trinomial. $\newline$ The distributive property states that $a(b + c + d) = ab + ac + ad$ . $\newline$ Applying this to our polynomials, we get: $\newline$ $(2z-1)(z^2) + (2z-1)(-2z) + (2z-1)(1)$
Distribute Terms: Distribute each term in the binomial across the terms in the trinomial. $\newline$ This gives us: $\newline$ $(2z \cdot z^2) + (2z \cdot -2z) + (2z \cdot 1) + (-1 \cdot z^2) + (-1 \cdot -2z) + (-1 \cdot 1)$
Perform Multiplication: Perform the multiplication for each term. $\newline$ Now we multiply the coefficients and add the exponents for terms with the same base: $\newline$ $(2z^3) + (-4z^2) + (2z) + (-z^2) + (2z) + (-1)$
Combine Like Terms: Combine like terms. $\newline$ We combine terms with the same exponents: $\newline$ $2z^3 + (-4z^2 - z^2) + (2z + 2z) - 1$ $\newline$ This simplifies to: $\newline$ $2z^3 - 5z^2 + 4z - 1$