1. **Stating the problem:** Calculate the following polynomial operations given: For the first set: $$A(x) = 4x^3 + 3x - 4, \quad B(x) = x^2 + x + 5$$ For the second set: $$A(x) = 3x^3 + 2x^2 - x + 6, \quad B(x) = 2x^3 - 1, \quad C(x) = -2x^2 + 3x + 1$$ --- 2. **First set calculations:** - $3B = 3(x^2 + x + 5) = 3x^2 + 3x + 15$ - $2A - B = 2(4x^3 + 3x - 4) - (x^2 + x + 5) = 8x^3 + 6x - 8 - x^2 - x - 5 = 8x^3 - x^2 + 5x - 13$ - $4A - B = 4(4x^3 + 3x - 4) - (x^2 + x + 5) = 16x^3 + 12x - 16 - x^2 - x - 5 = 16x^3 - x^2 + 11x - 21$ - $A \cdot B = (4x^3 + 3x - 4)(x^2 + x + 5)$ Multiply term by term: $$4x^3 \cdot x^2 = 4x^5$$ $$4x^3 \cdot x = 4x^4$$ $$4x^3 \cdot 5 = 20x^3$$ $$3x \cdot x^2 = 3x^3$$ $$3x \cdot x = 3x^2$$ $$3x \cdot 5 = 15x$$ $$-4 \cdot x^2 = -4x^2$$ $$-4 \cdot x = -4x$$ $$-4 \cdot 5 = -20$$ Sum all: $$4x^5 + 4x^4 + (20x^3 + 3x^3) + (3x^2 - 4x^2) + (15x - 4x) - 20 = 4x^5 + 4x^4 + 23x^3 - x^2 + 11x - 20$$ - $2A + 2B = 2(4x^3 + 3x - 4) + 2(x^2 + x + 5) = 8x^3 + 6x - 8 + 2x^2 + 2x + 10 = 8x^3 + 2x^2 + 8x + 2$ - $4AB = 4(4x^3 + 3x - 4)(x^2 + x + 5) = 4 \times (A \cdot B)$ from above Multiply $A \cdot B$ by 4: $$4 \times (4x^5 + 4x^4 + 23x^3 - x^2 + 11x - 20) = 16x^5 + 16x^4 + 92x^3 - 4x^2 + 44x - 80$$ - $B^3 = (x^2 + x + 5)^3$ First find $B^2$: $$(x^2 + x + 5)^2 = x^4 + 2x^3 + 11x^2 + 10x + 25$$ Then multiply $B^2$ by $B$: $$(x^4 + 2x^3 + 11x^2 + 10x + 25)(x^2 + x + 5)$$ Multiply term by term and sum: $$x^4 \cdot x^2 = x^6$$ $$x^4 \cdot x = x^5$$ $$x^4 \cdot 5 = 5x^4$$ $$2x^3 \cdot x^2 = 2x^5$$ $$2x^3 \cdot x = 2x^4$$ $$2x^3 \cdot 5 = 10x^3$$ $$11x^2 \cdot x^2 = 11x^4$$ $$11x^2 \cdot x = 11x^3$$ $$11x^2 \cdot 5 = 55x^2$$ $$10x \cdot x^2 = 10x^3$$ $$10x \cdot x = 10x^2$$ $$10x \cdot 5 = 50x$$ $$25 \cdot x^2 = 25x^2$$ $$25 \cdot x = 25x$$ $$25 \cdot 5 = 125$$ Sum all: $$x^6 + (x^5 + 2x^5) + (5x^4 + 2x^4 + 11x^4) + (10x^3 + 11x^3 + 10x^3) + (55x^2 + 10x^2 + 25x^2) + (50x + 25x) + 125$$ $$= x^6 + 3x^5 + 18x^4 + 31x^3 + 90x^2 + 75x + 125$$ --- 3. **Second set calculations:** Given: $$A = 3x^3 + 2x^2 - x + 6, \quad B = 2x^3 - 1, \quad C = -2x^2 + 3x + 1$$ - $A + B = (3x^3 + 2x^2 - x + 6) + (2x^3 - 1) = 5x^3 + 2x^2 - x + 5$ - $4A - 2C = 4(3x^3 + 2x^2 - x + 6) - 2(-2x^2 + 3x + 1) = 12x^3 + 8x^2 - 4x + 24 + 4x^2 - 6x - 2 = 12x^3 + 12x^2 - 10x + 22$ - $2B + 3C = 2(2x^3 - 1) + 3(-2x^2 + 3x + 1) = 4x^3 - 2 - 6x^2 + 9x + 3 = 4x^3 - 6x^2 + 9x + 1$ - $A + B + C = (3x^3 + 2x^2 - x + 6) + (2x^3 - 1) + (-2x^2 + 3x + 1) = 5x^3 + 0x^2 + 2x + 6$ - $2C \cdot B = 2(-2x^2 + 3x + 1)(2x^3 - 1)$ First multiply inside: $$2C = -4x^2 + 6x + 2$$ Multiply by $B$: $$(-4x^2 + 6x + 2)(2x^3 - 1)$$ Multiply term by term: $$-4x^2 \cdot 2x^3 = -8x^5$$ $$-4x^2 \cdot (-1) = 4x^2$$ $$6x \cdot 2x^3 = 12x^4$$ $$6x \cdot (-1) = -6x$$ $$2 \cdot 2x^3 = 4x^3$$ $$2 \cdot (-1) = -2$$ Sum all: $$-8x^5 + 12x^4 + 4x^3 + 4x^2 - 6x - 2$$ - $C^2 = (-2x^2 + 3x + 1)^2$ Square the polynomial: $$( -2x^2 )^2 = 4x^4$$ $$2 \times (-2x^2) \times 3x = -12x^3$$ $$2 \times (-2x^2) \times 1 = -4x^2$$ $$(3x)^2 = 9x^2$$ $$2 \times 3x \times 1 = 6x$$ $$1^2 = 1$$ Sum all: $$4x^4 - 12x^3 + ( -4x^2 + 9x^2 ) + 6x + 1 = 4x^4 - 12x^3 + 5x^2 + 6x + 1$$ --- **Final answers:** **First set:** - $3B = 3x^2 + 3x + 15$ - $2A - B = 8x^3 - x^2 + 5x - 13$ - $4A - B = 16x^3 - x^2 + 11x - 21$ - $A \cdot B = 4x^5 + 4x^4 + 23x^3 - x^2 + 11x - 20$ - $2A + 2B = 8x^3 + 2x^2 + 8x + 2$ - $4AB = 16x^5 + 16x^4 + 92x^3 - 4x^2 + 44x - 80$ - $B^3 = x^6 + 3x^5 + 18x^4 + 31x^3 + 90x^2 + 75x + 125$ **Second set:** - $A + B = 5x^3 + 2x^2 - x + 5$ - $4A - 2C = 12x^3 + 12x^2 - 10x + 22$ - $2B + 3C = 4x^3 - 6x^2 + 9x + 1$ - $A + B + C = 5x^3 + 2x + 6$ - $2C \cdot B = -8x^5 + 12x^4 + 4x^3 + 4x^2 - 6x - 2$ - $C^2 = 4x^4 - 12x^3 + 5x^2 + 6x + 1$