1. **Problem 25:** Determine which of the options is NOT a factor of the trinomial $24x^3 - 6x^2 - 9x$. 2. **Step 1:** Factor out the greatest common factor (GCF) from the trinomial: $$24x^3 - 6x^2 - 9x = 3x(8x^2 - 2x - 3)$$ 3. **Step 2:** Factor the quadratic inside the parentheses: We look for two numbers that multiply to $8 \times (-3) = -24$ and add to $-2$. These numbers are $-6$ and $4$. 4. **Step 3:** Rewrite and factor by grouping: $$8x^2 - 2x - 3 = 8x^2 - 6x + 4x - 3 = 2x(4x - 3) + 1(4x - 3) = (2x + 1)(4x - 3)$$ 5. **Step 4:** So the full factorization is: $$3x(2x + 1)(4x - 3)$$ 6. **Step 5:** Check each option: - A. $3x$ is a factor. - B. $4x - 3$ is a factor. - C. $2x + 1$ is a factor. - D. $2x - 1$ is NOT a factor. **Answer for 25:** D. $2x - 1$ --- 1. **Problem 26:** Find which binomial is a factor of $4x^2 + 12x + 5$. 2. **Step 1:** Factor the quadratic $4x^2 + 12x + 5$. 3. **Step 2:** Multiply $4 \times 5 = 20$, find two numbers that multiply to 20 and add to 12: 10 and 2. 4. **Step 3:** Rewrite and factor by grouping: $$4x^2 + 10x + 2x + 5 = 2x(2x + 5) + 1(2x + 5) = (2x + 1)(2x + 5)$$ 5. **Step 4:** Check options: - A. $2x + 5$ is a factor. - B. $2x - 5$ is not. - C. $4x + 1$ is not. - D. $4x - 1$ is not. **Answer for 26:** A. $2x + 5$ --- 1. **Problem 27:** Given volume $V = 6x^3 + 3x^2 - 18x$, find possible length, width, height. 2. **Step 1:** Factor out GCF: $$6x^3 + 3x^2 - 18x = 3x(2x^2 + x - 6)$$ 3. **Step 2:** Factor quadratic: Find two numbers multiplying to $2 \times (-6) = -12$ and adding to $1$: 4 and -3. 4. **Step 3:** Factor by grouping: $$2x^2 + 4x - 3x - 6 = 2x(x + 2) - 3(x + 2) = (2x - 3)(x + 2)$$ 5. **Step 4:** So volume factors as: $$3x(2x - 3)(x + 2)$$ 6. **Step 5:** Possible dimensions: Length = $3x$, Width = $2x - 3$, Height = $x + 2$ (or any permutation) --- 1. **Problem 28:** Given volume $V = 10x^3 - 55x + 60$, find possible length, width, height. 2. **Step 1:** Factor out GCF: $$10x^3 - 55x + 60 = 5(2x^3 - 11x + 12)$$ 3. **Step 2:** Factor cubic $2x^3 - 11x + 12$ by trial of rational roots. Try $x=3$: $$2(3)^3 - 11(3) + 12 = 54 - 33 + 12 = 33 eq 0$$ Try $x=2$: $$2(2)^3 - 11(2) + 12 = 16 - 22 + 12 = 6 eq 0$$ Try $x=1$: $$2(1)^3 - 11(1) + 12 = 2 - 11 + 12 = 3 eq 0$$ Try $x=-1$: $$2(-1)^3 - 11(-1) + 12 = -2 + 11 + 12 = 21 eq 0$$ Try $x=4$: $$2(4)^3 - 11(4) + 12 = 128 - 44 + 12 = 96 eq 0$$ Try $x=-3$: $$2(-3)^3 - 11(-3) + 12 = -54 + 33 + 12 = -9 eq 0$$ 4. **Step 3:** Use synthetic division or factor by grouping: Try to factor as $(x - a)(2x^2 + bx + c)$. 5. **Step 4:** Alternatively, factor by grouping: Rewrite as $2x^3 - 11x + 12 = 2x^3 + 0x^2 - 11x + 12$ 6. **Step 5:** Try to factor as $(2x - 3)(x^2 + px + q)$: Expand: $$2x^3 + 2px^2 + 2qx - 3x^2 - 3px - 3q = 2x^3 + (2p - 3)x^2 + (2q - 3p)x - 3q$$ 7. **Step 6:** Match coefficients: - Coefficient of $x^3$: $2$ (matches) - Coefficient of $x^2$: $0 = 2p - 3$ so $2p = 3$ so $p = \frac{3}{2}$ - Coefficient of $x$: $-11 = 2q - 3p = 2q - 3 \times \frac{3}{2} = 2q - \frac{9}{2}$ So $2q = -11 + \frac{9}{2} = -\frac{22}{2} + \frac{9}{2} = -\frac{13}{2}$, so $q = -\frac{13}{4}$ - Constant term: $12 = -3q = -3 \times -\frac{13}{4} = \frac{39}{4}$ which is not equal to 12. 8. **Step 7:** So $(2x - 3)$ is not a factor. 9. **Step 8:** Try $x - 3$: Divide $2x^3 - 11x + 12$ by $x - 3$ using synthetic division: Coefficients: 2, 0, -11, 12 Bring down 2 Multiply 2*3=6, add to 0=6 Multiply 6*3=18, add to -11=7 Multiply 7*3=21, add to 12=33 (not zero remainder) 10. **Step 9:** Try $x + 3$: Synthetic division with -3: Bring down 2 Multiply 2*(-3)=-6, add to 0=-6 Multiply -6*(-3)=18, add to -11=7 Multiply 7*(-3)=-21, add to 12=-9 (not zero) 11. **Step 10:** Try $x - 2$: Synthetic division with 2: Bring down 2 Multiply 2*2=4, add to 0=4 Multiply 4*2=8, add to -11=-3 Multiply -3*2=-6, add to 12=6 (not zero) 12. **Step 11:** Try $x + 2$: Synthetic division with -2: Bring down 2 Multiply 2*(-2)=-4, add to 0=-4 Multiply -4*(-2)=8, add to -11=-3 Multiply -3*(-2)=6, add to 12=18 (not zero) 13. **Step 12:** Try $x - 1$: Synthetic division with 1: Bring down 2 Multiply 2*1=2, add to 0=2 Multiply 2*1=2, add to -11=-9 Multiply -9*1=-9, add to 12=3 (not zero) 14. **Step 13:** Try $x + 1$: Synthetic division with -1: Bring down 2 Multiply 2*(-1)=-2, add to 0=-2 Multiply -2*(-1)=2, add to -11=-9 Multiply -9*(-1)=9, add to 12=21 (not zero) 15. **Step 14:** Since no rational roots, try factoring quadratic part of options: Check options: - A. $2x + 5$ - B. $2x - 5$ - C. $4x + 1$ - D. $4x - 1$ 16. **Step 15:** Test each by polynomial division or substitution: Try dividing $10x^3 - 55x + 60$ by each binomial: 17. **Step 16:** Divide by $2x - 5$: Use synthetic division with root $\frac{5}{2} = 2.5$: Coefficients: 10, 0, -55, 60 Bring down 10 Multiply 10*2.5=25, add to 0=25 Multiply 25*2.5=62.5, add to -55=7.5 Multiply 7.5*2.5=18.75, add to 60=78.75 (not zero) 18. **Step 17:** Divide by $2x + 5$ (root $-2.5$): Bring down 10 Multiply 10*(-2.5)=-25, add to 0=-25 Multiply -25*(-2.5)=62.5, add to -55=7.5 Multiply 7.5*(-2.5)=-18.75, add to 60=41.25 (not zero) 19. **Step 18:** Divide by $4x - 1$ (root $\frac{1}{4} = 0.25$): Bring down 10 Multiply 10*0.25=2.5, add to 0=2.5 Multiply 2.5*0.25=0.625, add to -55=-54.375 Multiply -54.375*0.25=-13.59375, add to 60=46.40625 (not zero) 20. **Step 19:** Divide by $4x + 1$ (root $-0.25$): Bring down 10 Multiply 10*(-0.25)=-2.5, add to 0=-2.5 Multiply -2.5*(-0.25)=0.625, add to -55=-54.375 Multiply -54.375*(-0.25)=13.59375, add to 60=73.59375 (not zero) 21. **Step 20:** None of the options are factors of the cubic polynomial. 22. **Step 21:** Since the problem asks for possible expressions for length, width, and height, and the volume is given as $10x^3 - 55x + 60$, factor out GCF 5: $$5(2x^3 - 11x + 12)$$ 23. **Step 22:** Use rational root theorem to find roots of $2x^3 - 11x + 12$. Try $x=3$: $$2(3)^3 - 11(3) + 12 = 54 - 33 + 12 = 33 \neq 0$$ Try $x= -2$: $$2(-2)^3 - 11(-2) + 12 = -16 + 22 + 12 = 18 \neq 0$$ Try $x= 4$: $$2(4)^3 - 11(4) + 12 = 128 - 44 + 12 = 96 \neq 0$$ 24. **Step 23:** Since no rational roots, the cubic is irreducible over rationals. 25. **Step 24:** So possible expressions for length, width, height could be: Length = $5$ Width = $x$ Height = $2x^2 - 11 + \frac{12}{x^2}$ (or other algebraic expressions) **Answer for 28:** Cannot factor nicely; length, width, height expressions are not simple binomials. --- **Summary:** - Problem 25 answer: D - Problem 26 answer: A - Problem 27 factors: $3x$, $2x - 3$, $x + 2$ - Problem 28: No simple binomial factors; volume factored as $5(2x^3 - 11x + 12)$