It is given that 10x^(3)+5x^(2)-x+4=(2x-1)(x+3)Q(x)+Ax+B, where Q(x) is a polynomial. Find the values of the constants A and B.

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Understand Polynomial Division: Understand the division of polynomials. The given equation is a polynomial division where the dividend is $10x^3 + 5x^2 - x + 4$ and the divisor is $(2x - 1)(x + 3)$. The quotient is $Q(x)$ and the remainder is $Ax + B$. To find $A$ and $B$, we need to perform polynomial long division or synthetic division.Set Up Division: Set up the division. Since $Q(x)$ is a polynomial and we are given that the remainder is linear (of the form $Ax + B$), we can infer that $Q(x)$ must be of degree 1 (since the degree of the dividend is 3 and the divisor is 2). Let's denote $Q(x) = Cx + D$ where $C$ and $D$ are constants to be determined.Multiply Divisor by Quotient: Multiply the divisor by the assumed quotient. We multiply $(2x - 1)(x + 3)$ by $Q(x) = Cx + D$ to get the part of the dividend that is divisible by the divisor. $(2x - 1)(x + 3)(Cx + D) = (2Cx^2 + (6C - D)x - D)(x + 3)$ Now we expand this product to find the coefficients that will match the dividend.Expand and Equate Coefficients: Expand the product and equate coefficients. Expanding the product, we get: $2Cx^3 + (6C - D)x^2 - Dx + 2Cx^2 + (6C - D)x - D$ Combining like terms, we have: $2Cx^3 + (6C - D + 2C)x^2 + (6C - D - D)x - D$ This simplifies to: $2Cx^3 + (8C - D)x^2 + (6C - 2D)x - D$ Now we equate the coefficients of this expression to the coefficients of the dividend $10x^3 + 5x^2 - x + 4$.Equate x^3 Coefficients: Equate the coefficients of the x^3 terms. From the x^3 terms, we have: $2C = 10$ Solving for $C$, we get: $C = 5$Equate x^2 Coefficients: Equate the coefficients of the x^2 terms. From the x^2 terms, we have: $8C - D = 5$ Substituting $C = 5$ into this equation, we get: $8(5) - D = 5$ Solving for $D$, we get: $40 - D = 5$ $D = 40 - 5$ $D = 35$Realize and Correct Mistake: Realize the mistake and correct it. We made a mistake in the previous step. We should have equated the coefficients of the x^2 terms to the dividend's x^2 coefficient, which is 5. However, we incorrectly equated $8C - D$ to 5 instead of $8C - D$ to the dividend's x^2 coefficient. Let's correct this.

It is given that $10 x^{3}+5 x^{2}-x+4=(2 x-1)(x+3) \mathrm{Q}(x)+A x+B$ , where $\mathrm{Q}(x)$ is a polynomial. Find the values of the constants $A$ and $B$ .

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It is given that 10x3+5x2−x+4=(2x−1)(x+3)Q(x)+Ax+B 10 x^{3}+5 x^{2}-x+4=(2 x-1)(x+3) \mathrm{Q}(x)+A x+B 10x3+5x2−x+4=(2x−1)(x+3)Q(x)+Ax+B, where Q(x) \mathrm{Q}(x) Q(x) is a polynomial. Find the values of the constants A A A and B B B.

It is given that $10 x^{3}+5 x^{2}-x+4=(2 x-1)(x+3) \mathrm{Q}(x)+A x+B$ , where $\mathrm{Q}(x)$ is a polynomial. Find the values of the constants $A$ and $B$ .