1. **Problem:** Find all possible values of $b$ for which $7x^2 + bx + 3$ is factorable into linear factors with integer coefficients and constants. 2. **Formula and rules:** For a quadratic $ax^2 + bx + c$ to factor into $(mx + n)(px + q)$ with integers $m,n,p,q$, the product $ac$ must have factor pairs whose sums equal $b$ when combined appropriately. 3. Here, $a=7$, $c=3$, so $ac=21$. 4. Factor pairs of 21 are $(1,21)$ and $(3,7)$, with their negatives. 5. Since $a=7$ is prime, possible factor pairs for the factors are $(7x + n)(x + q)$ or $(x + n)(7x + q)$. 6. Let the factors be $(7x + m)(x + n) = 7x^2 + (7n + m)x + mn$. 7. We want $mn = 3$ and $7n + m = b$. 8. Possible integer pairs $(m,n)$ with $mn=3$ are $(1,3)$, $(3,1)$, $(-1,-3)$, $(-3,-1)$. 9. Calculate $b=7n + m$ for each: - $(1,3)$: $b=7*3 + 1=22$ - $(3,1)$: $b=7*1 + 3=10$ - $(-1,-3)$: $b=7*(-3) + (-1)=-22$ - $(-3,-1)$: $b=7*(-1) + (-3)=-10$ 10. So possible $b$ values are $22, 10, -22, -10$. --- 1. **Problem:** Can $3x^2 + 5x + 3$ be factored into linear factors with integer coefficients? 2. **Check:** $a=3$, $c=3$, so $ac=9$. 3. Factor pairs of 9: $(1,9)$, $(3,3)$, and their negatives. 4. Try to find integers $m,n$ such that $mn=3$ and $3n + m = 5$ (assuming factors $(3x + m)(x + n)$). 5. Possible $(m,n)$ with $mn=3$: $(1,3)$, $(3,1)$, $(-1,-3)$, $(-3,-1)$. 6. Calculate $3n + m$: - $(1,3)$: $3*3 + 1=10$ - $(3,1)$: $3*1 + 3=6$ - $(-1,-3)$: $3*(-3) + (-1)=-10$ - $(-3,-1)$: $3*(-1) + (-3)=-6$ 7. None equals 5, so no integer factorization. --- 1. **Problem:** Correct the error in factoring $2x^2 + 11x + 15$. 2. Given $ac=30$, $b=11$. 3. Factor pairs of 30 and their sums: - $1,30$ sum 31 - $2,15$ sum 17 - $3,10$ sum 13 - $5,6$ sum 11 4. Correct factorization uses $5$ and $6$. 5. Rewrite middle term: $2x^2 + 5x + 6x + 15$. 6. Group: $(2x^2 + 5x) + (6x + 15)$. 7. Factor each: $x(2x + 5) + 3(2x + 5)$. 8. Factor out common binomial: $(x + 3)(2x + 5)$. 9. The student's error was writing $(x + 5)(x + 6)$ which is incorrect. --- 1. **Problem:** Factor $6x^2 + 7x - 6$ into $(px + q)(sx + t)$ with integers. 2. $a=6$, $c=-6$, so $ac=-36$. 3. Factor pairs of $-36$ with sums equal to $b=7$. 4. Possible pairs: $(9,-4)$ sum 5, $(12,-3)$ sum 9, $(18,-2)$ sum 16, $(6,-6)$ sum 0, $(3,-12)$ sum -9, $(4,-9)$ sum -5. 5. None sum to 7. 6. Try $(px + q)(sx + t)$ with $p*s=6$ and $q*t=-6$. 7. Possible $(p,s)$: $(1,6)$, $(2,3)$, $(3,2)$, $(6,1)$. 8. Test each with $q,t$ such that $q*t=-6$ and $p*t + q*s = 7$. 9. For $(2,3)$: - Try $q=6$, $t=-1$: $2*(-1) + 6*3 = -2 + 18 = 16$ - Try $q=3$, $t=-2$: $2*(-2) + 3*3 = -4 + 9 = 5$ - Try $q=-3$, $t=2$: $2*2 + (-3)*3 = 4 - 9 = -5$ - Try $q=-6$, $t=1$: $2*1 + (-6)*3 = 2 - 18 = -16$ 10. For $(3,2)$: - $q=6$, $t=-1$: $3*(-1) + 6*2 = -3 + 12 = 9$ - $q=3$, $t=-2$: $3*(-2) + 3*2 = -6 + 6 = 0$ - $q=-3$, $t=2$: $3*2 + (-3)*2 = 6 - 6 = 0$ - $q=-6$, $t=1$: $3*1 + (-6)*2 = 3 - 12 = -9$ 11. For $(1,6)$: - $q=6$, $t=-1$: $1*(-1) + 6*6 = -1 + 36 = 35$ - $q=3$, $t=-2$: $1*(-2) + 3*6 = -2 + 18 = 16$ - $q=-3$, $t=2$: $1*2 + (-3)*6 = 2 - 18 = -16$ - $q=-6$, $t=1$: $1*1 + (-6)*6 = 1 - 36 = -35$ 12. For $(6,1)$: - $q=6$, $t=-1$: $6*(-1) + 6*1 = -6 + 6 = 0$ - $q=3$, $t=-2$: $6*(-2) + 3*1 = -12 + 3 = -9$ - $q=-3$, $t=2$: $6*2 + (-3)*1 = 12 - 3 = 9$ - $q=-6$, $t=1$: $6*1 + (-6)*1 = 6 - 6 = 0$ 13. None equal 7, so $6x^2 + 7x - 6$ cannot be factored into linear factors with integer coefficients. --- 1. **Problem:** Factor $4x^2 + 16x + 12$. 2. Factor out common factor 4: $4(x^2 + 4x + 3)$. 3. Factor inside: $x^2 + 4x + 3 = (x + 3)(x + 1)$. 4. Final factorization: $4(x + 3)(x + 1)$. --- 1. **Problem:** Factor $2x^2 - 16x + 30$. 2. Factor out 2: $2(x^2 - 8x + 15)$. 3. Factor inside: $x^2 - 8x + 15 = (x - 3)(x - 5)$. 4. Final factorization: $2(x - 3)(x - 5)$. --- 1. **Problem:** Factor $3x^2 + 12x - 63$. 2. Factor out 3: $3(x^2 + 4x - 21)$. 3. Factor inside: $x^2 + 4x - 21 = (x + 7)(x - 3)$. 4. Final factorization: $3(x + 7)(x - 3)$. --- 1. **Problem:** Factor $6x^2 + 12x - 48$. 2. Factor out 6: $6(x^2 + 2x - 8)$. 3. Factor inside: $x^2 + 2x - 8 = (x + 4)(x - 2)$. 4. Final factorization: $6(x + 4)(x - 2)$. --- 1. **Problem:** Factor $7x^2 + 9x + 2$. 2. $ac = 7*2 = 14$. 3. Find factor pairs of 14 that sum to 9: $(7,2)$ sum 9. 4. Rewrite middle term: $7x^2 + 7x + 2x + 2$. 5. Group: $(7x^2 + 7x) + (2x + 2)$. 6. Factor each: $7x(x + 1) + 2(x + 1)$. 7. Factor out common binomial: $(7x + 2)(x + 1)$. --- 1. **Problem:** Factor $6x^2 + 11x - 2$. 2. $ac = 6*(-2) = -12$. 3. Factor pairs of -12 that sum to 11: $(12, -1)$ sum 11. 4. Rewrite middle term: $6x^2 + 12x - x - 2$. 5. Group: $(6x^2 + 12x) + (-x - 2)$. 6. Factor each: $6x(x + 2) -1(x + 2)$. 7. Factor out common binomial: $(6x - 1)(x + 2)$. --- 1. **Problem:** Factor $8x^2 - 2x - 1$. 2. $ac = 8*(-1) = -8$. 3. Factor pairs of -8 that sum to -2: $(2, -4)$ sum -2. 4. Rewrite middle term: $8x^2 + 2x - 4x - 1$. 5. Group: $(8x^2 + 2x) + (-4x - 1)$. 6. Factor each: $2x(4x + 1) -1(4x + 1)$. 7. Factor out common binomial: $(2x - 1)(4x + 1)$. --- 1. **Problem:** Factor $10x^2 + 19x + 6$. 2. $ac = 10*6 = 60$. 3. Factor pairs of 60 that sum to 19: $(15,4)$ sum 19. 4. Rewrite middle term: $10x^2 + 15x + 4x + 6$. 5. Group: $(10x^2 + 15x) + (4x + 6)$. 6. Factor each: $5x(2x + 3) + 2(2x + 3)$. 7. Factor out common binomial: $(5x + 2)(2x + 3)$. --- 1. **Problem:** Factor $15x^2 - 16x - 7$. 2. $ac = 15*(-7) = -105$. 3. Factor pairs of -105 that sum to -16: $(-21,5)$ sum -16. 4. Rewrite middle term: $15x^2 - 21x + 5x - 7$. 5. Group: $(15x^2 - 21x) + (5x - 7)$. 6. Factor each: $3x(5x - 7) + 1(5x - 7)$. 7. Factor out common binomial: $(3x + 1)(5x - 7)$. --- 1. **Problem:** Factor $12x^2 + 11x + 2$. 2. $ac = 12*2 = 24$. 3. Factor pairs of 24 that sum to 11: $(8,3)$ sum 11. 4. Rewrite middle term: $12x^2 + 8x + 3x + 2$. 5. Group: $(12x^2 + 8x) + (3x + 2)$. 6. Factor each: $4x(3x + 2) + 1(3x + 2)$. 7. Factor out common binomial: $(4x + 1)(3x + 2)$. --- 1. **Problem:** Factor $4x^2 + 13x + 3$. 2. $ac = 4*3 = 12$. 3. Factor pairs of 12 that sum to 13: $(12,1)$ sum 13. 4. Rewrite middle term: $4x^2 + 12x + x + 3$. 5. Group: $(4x^2 + 12x) + (x + 3)$. 6. Factor each: $4x(x + 3) + 1(x + 3)$. 7. Factor out common binomial: $(4x + 1)(x + 3)$. --- 1. **Problem:** Factor $6x^2 - 25x - 14$. 2. $ac = 6*(-14) = -84$. 3. Factor pairs of -84 that sum to -25: $(-28,3)$ sum -25. 4. Rewrite middle term: $6x^2 - 28x + 3x - 14$. 5. Group: $(6x^2 - 28x) + (3x - 14)$. 6. Factor each: $2x(3x - 14) + 1(3x - 14)$. 7. Factor out common binomial: $(2x + 1)(3x - 14)$. --- 1. **Problem:** Factor $2x^2 + 7x - 4$. 2. $ac = 2*(-4) = -8$. 3. Factor pairs of -8 that sum to 7: $(8, -1)$ sum 7. 4. Rewrite middle term: $2x^2 + 8x - x - 4$. 5. Group: $(2x^2 + 8x) + (-x - 4)$. 6. Factor each: $2x(x + 4) -1(x + 4)$. 7. Factor out common binomial: $(2x - 1)(x + 4)$. --- 1. **Problem:** Factor $12x^2 + 13x + 3$. 2. $ac = 12*3 = 36$. 3. Factor pairs of 36 that sum to 13: $(9,4)$ sum 13. 4. Rewrite middle term: $12x^2 + 9x + 4x + 3$. 5. Group: $(12x^2 + 9x) + (4x + 3)$. 6. Factor each: $3x(4x + 3) + 1(4x + 3)$. 7. Factor out common binomial: $(3x + 1)(4x + 3)$. --- 1. **Problem:** Factor $6x^3 + 9x^2 + 3x$. 2. Factor out $3x$: $3x(2x^2 + 3x + 1)$. 3. Factor inside quadratic: $2x^2 + 3x + 1 = (2x + 1)(x + 1)$. 4. Final factorization: $3x(2x + 1)(x + 1)$. --- 1. **Problem:** Factor $8x^2 - 10x - 3$. 2. $ac = 8*(-3) = -24$. 3. Factor pairs of -24 that sum to -10: $(-12, 2)$ sum -10. 4. Rewrite middle term: $8x^2 - 12x + 2x - 3$. 5. Group: $(8x^2 - 12x) + (2x - 3)$. 6. Factor each: $4x(2x - 3) + 1(2x - 3)$. 7. Factor out common binomial: $(4x + 1)(2x - 3)$. --- 1. **Problem:** Factor $12x^2 + 16x + 5$. 2. $ac = 12*5 = 60$. 3. Factor pairs of 60 that sum to 16: $(10,6)$ sum 16. 4. Rewrite middle term: $12x^2 + 10x + 6x + 5$. 5. Group: $(12x^2 + 10x) + (6x + 5)$. 6. Factor each: $2x(6x + 5) + 1(6x + 5)$. 7. Factor out common binomial: $(2x + 1)(6x + 5)$. --- 1. **Problem:** Factor $16x^3 + 32x^2 + 12x$. 2. Factor out $4x$: $4x(4x^2 + 8x + 3)$. 3. Factor inside quadratic: $4x^2 + 8x + 3$. 4. $ac = 4*3=12$, factor pairs summing to 8: $(6,2)$ sum 8. 5. Rewrite middle term: $4x^2 + 6x + 2x + 3$. 6. Group: $(4x^2 + 6x) + (2x + 3)$. 7. Factor each: $2x(2x + 3) + 1(2x + 3)$. 8. Factor out common binomial: $(2x + 1)(2x + 3)$. 9. Final factorization: $4x(2x + 1)(2x + 3)$. --- 1. **Problem:** Factor $21x^2 - 35x - 14$. 2. Factor out 7: $7(3x^2 - 5x - 2)$. 3. $ac = 3*(-2) = -6$. 4. Factor pairs of -6 that sum to -5: $(-6,1)$ sum -5. 5. Rewrite middle term: $3x^2 - 6x + x - 2$. 6. Group: $(3x^2 - 6x) + (x - 2)$. 7. Factor each: $3x(x - 2) + 1(x - 2)$. 8. Factor out common binomial: $(3x + 1)(x - 2)$. 9. Final factorization: $7(3x + 1)(x - 2)$. --- 1. **Problem:** Factor $16x^2 + 22x - 3$. 2. $ac = 16*(-3) = -48$. 3. Factor pairs of -48 that sum to 22: $(24, -2)$ sum 22. 4. Rewrite middle term: $16x^2 + 24x - 2x - 3$. 5. Group: $(16x^2 + 24x) + (-2x - 3)$. 6. Factor each: $8x(2x + 3) -1(2x + 3)$. 7. Factor out common binomial: $(8x - 1)(2x + 3)$. --- 1. **Problem:** Factor $9x^2 + 46x + 5$. 2. $ac = 9*5 = 45$. 3. Factor pairs of 45 that sum to 46: $(45,1)$ sum 46. 4. Rewrite middle term: $9x^2 + 45x + x + 5$. 5. Group: $(9x^2 + 45x) + (x + 5)$. 6. Factor each: $9x(x + 5) + 1(x + 5)$. 7. Factor out common binomial: $(9x + 1)(x + 5)$. --- 1. **Problem:** Factor $24x^3 - 10x^2 - 4x$. 2. Factor out $2x$: $2x(12x^2 - 5x - 2)$. 3. $ac = 12*(-2) = -24$. 4. Factor pairs of -24 that sum to -5: $(-8,3)$ sum -5. 5. Rewrite middle term: $12x^2 - 8x + 3x - 2$. 6. Group: $(12x^2 - 8x) + (3x - 2)$. 7. Factor each: $4x(3x - 2) + 1(3x - 2)$. 8. Factor out common binomial: $(4x + 1)(3x - 2)$. 9. Final factorization: $2x(4x + 1)(3x - 2)$. --- 1. **Problem:** Factor $3x^2 + xy - 2y^2$. 2. Treat as quadratic in $x$: $a=3$, $b=y$, $c=-2y^2$. 3. $ac = 3*(-2y^2) = -6y^2$. 4. Factor pairs of $-6y^2$ that sum to $y$: $(3y, -2y)$ sum $y$. 5. Rewrite middle term: $3x^2 + 3xy - 2xy - 2y^2$. 6. Group: $(3x^2 + 3xy) + (-2xy - 2y^2)$. 7. Factor each: $3x(x + y) - 2y(x + y)$. 8. Factor out common binomial: $(3x - 2y)(x + y)$. --- 1. **Problem:** Factor $2x^2 + 9xy + 10y^2$. 2. $a=2$, $b=9y$, $c=10y^2$. 3. $ac = 2*10y^2 = 20y^2$. 4. Factor pairs of $20y^2$ that sum to $9y$: $(5y, 4y)$ sum $9y$. 5. Rewrite middle term: $2x^2 + 5xy + 4xy + 10y^2$. 6. Group: $(2x^2 + 5xy) + (4xy + 10y^2)$. 7. Factor each: $x(2x + 5y) + 2y(2x + 5y)$. 8. Factor out common binomial: $(x + 2y)(2x + 5y)$. --- 1. **Problem:** Factor $5x^2 - 4xy - y^2$. 2. $a=5$, $b=-4y$, $c=-y^2$. 3. $ac = 5*(-1)y^2 = -5y^2$. 4. Factor pairs of $-5y^2$ that sum to $-4y$: $(-5y, y)$ sum $-4y$. 5. Rewrite middle term: $5x^2 - 5xy + xy - y^2$. 6. Group: $(5x^2 - 5xy) + (xy - y^2)$. 7. Factor each: $5x(x - y) + y(x - y)$. 8. Factor out common binomial: $(5x + y)(x - y)$. --- 1. **Problem:** Factor $2x^2 + 10xy + 12y^2$. 2. Factor out 2: $2(x^2 + 5xy + 6y^2)$. 3. $a=1$, $b=5y$, $c=6y^2$. 4. Factor pairs of $6y^2$ that sum to $5y$: $(2y, 3y)$ sum $5y$. 5. Factor inside: $(x + 2y)(x + 3y)$. 6. Final factorization: $2(x + 2y)(x + 3y)$.